As a result, the angles with a common reference angle of 150° in each quadrant are as follows: 210° in the first quadrant, 330° in the second, 30° in the third, and 150° in the fourth (which is equivalent to 510°).
What does a math angle mean?An angle is created by combining a set of beams (half-lines) with the same shared terminal. The angle's vertex is the latter, while the rays are alternately referred to as both the angle's leg and its arms.
By deducting 210° from 360° and obtaining the exact value of the result, one may get the reference angle:
Angle of reference = |360° - 210°| = 150°
We may add or subtract multiples of 360° or 180° as necessary to determine angles in each quadrant with such a common reference angle of 210°.
First Quadrant:
By deducting 150 degrees from 360 degrees, one can find an angle for the first quadrant with such a reference angle of 150 degrees:
θ = 360° - 150° = 210°
Second Quadrant:
An angle in the second quadrant with a reference angle of 150° is found by adding 150° to 180°:
θ = 180° + 150° = 330°
Third Quadrant:
By deducting 150 degrees from 180 degrees, one can find an angle inside the third quadrant using a reference angle of 150 degrees:
θ = 180° - 150° = 30°
Fourth Quadrant:
Addition of 150 degrees to 360 degrees yields an angle inside the fourth quadrant with such a reference angle of 150 degrees:
θ = 360° + 150° = 510°
However, since we are looking for angles within the range of 0° to 360°, we can subtract 360° from the angle in the fourth quadrant to get an equivalent angle in the first quadrant:
θ = 510° - 360° = 150°
With such a common reference angle of 150°, the angles in each quadrant are as follows:
210° in the first quadrant
330° in the second quadrant
30° in the third quadrant
150° in the fourth quadrant (which is equivalent to 510°
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When expressions of the form (x −r)(x − s) are multiplied out, a quadratic polynomial is obtained. For instance, (x −2)(x −(−7))= (x −2)(x + 7) = x2 + 5x − 14.
a. What can be said about the coefficients of the polynomial obtained by multiplying out (x −r)(x − s) when both r and s are odd integers? when both r and s are even integers? when one of r and s is even and the other is odd?
b. It follows from part (a) that x2 − 1253x + 255 cannot be written as a product of two polynomials with integer coefficients. Explain why this is so.
a.(1) When both r and s are odd integers, the quadratic polynomial obtained by multiplying out (x - r)(x - s) will have a coefficient of 1 for x^2 term, and both the coefficient of x term and constant term will be odd integers.
(2) When both r and s are even integers, the polynomial obtained by multiplying out (x - r)(x - s) will also have a coefficient of 1 for x^2 term, but the coefficient of x term and constant term will be even integers.
(3) When one of r and s is even and the other is odd, the polynomial obtained by multiplying out (x - r)(x - s) will have a coefficient of 1 for x^2 term, the coefficient of x term will be an odd integer, while the constant term will be an even integer.
b. x^2 - 1253x + 255 cannot be written as a product of two polynomials with integer coefficients.
a. When both r and s are odd integers, the product (x − r)(x − s) will have a coefficient of 1 for x^2 term, and both the coefficient of x term and constant term will be odd integers. This is because the sum of two odd integers and the product of two odd integers is also an odd integer.
When both r and s are even integers, the product (x − r)(x − s) will also have a coefficient of 1 for x^2 term, but the coefficient of x term and constant term will be even integers. This is because the sum of two even integers and the product of two even integers is also an even integer.
When one of r and s is even and the other is odd, the product (x − r)(x − s) will have a coefficient of 1 for x^2 term, and the coefficient of x term will be an odd integer, while the constant term will be an even integer. This is because the sum of an odd and even integer is an odd integer, and the product of an odd and even integer is an even integer.
b. If x^2 - 1253x + 255 can be written as a product of two polynomials with integer coefficients, then we can write it as (x - r)(x - s) where r and s are integers. From part (a), we know that both r and s cannot be odd integers since the coefficient of x term would be odd, but 1253 is an odd integer. Similarly, both r and s cannot be even integers since the constant term would be even, but 255 is an odd integer. Therefore, one of r and s must be odd and the other must be even. However, the difference between an odd integer and an even integer is always odd, so the coefficient of x term in the product (x - r)(x - s) would be odd, which is not equal to the coefficient of x term in x^2 - 1253x + 255. Hence, x^2 - 1253x + 255 cannot be written as a product of two polynomials with integer coefficients.
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John is standing on top of a cliff 275 feet above the ocean. The measuremment of the angle of depression to a boat in the ocean is 38 degrees. How far is the boat from the base of the cliff?
Answer: The boat is approximately 357.4 feet from the base of the cliff.
Step-by-step explanation:
Let x be the horizontal distance from the base of the cliff to the boat. Using the tangent function, we can write:
tan(38) = 275 / x
Solving for x, we have:
x = 275 / tan(38)
Using a calculator, we get:
x ≈ 357.4 feet
Therefore, the boat is approximately 357.4 feet from the base of the cliff.
Answer:
352m
Step-by-step explanation:
h = 275m
a = b (alternative angles)
.: b = 38°
Let the base from the boat to the cliff be d
Using TanTan 38° = opposite ÷ adjacent
Tan 38° ° = 275 ÷d
d = 275 ÷ Tan 38 °
d = 352m
.: The boat is 352m away from the foot of the cliff
A set of sweater prices are normally distributed with a mean of
58
5858 dollars and a standard deviation of
5
55 dollars.
What proportion of sweater prices are between
48.50
48.5048, point, 50 dollars and
60
6060 dollars?
Answer:
0.6267
Step-by-step explanation:
See the picture.
Hope its clear.
An individual is baking 3 batches of cookies. They used 1.8 oz. of vanilla in one batch of the cookies, 1.25 oz. of vanilla in the second batch and .95 oz. in the third batch. Convert these decimals into fractions, and then put them in ascending order.
Answer:
19/20 , 1 1/4 , 1 4/5
Step-by-step explanation:
1.8 = 1 4/5 (fraction)
1.8 converts to 18/10. This can be simplified twice, firstly by making it 9/5 since both 18 and 10 are divisible by two, but can be simplified further to 1 4/5
1.25 = 1 1/4 (fraction)
1.25 converts to 125/100. This can be simplified to 5/4 or 1 1/4
0.95 = 19/20 (fraction)
0.95 converts to 95/100. This can be simplified to 19/20
Ascending Order (smallest to largest)
smallest - 19/20
middle - 1 1/4
largest - 1 4/5
I believe this is the right answer, but haven't done fractions in a while so may want to double check to make sure
During a manufacturing process, a metal part in a machine is exposed to varying temperature conditions. The manufacturer of the machine recommends that the temperature of the machine part remain below 131°F. The temperature T in degrees Fahrenheit x minutes after the machine is put into operation is modeled by T=-0.005x^2+0.45x+125. Will the temperature of the part ever reach or exceed 131°F? Use the discriminant of a quadratic equation to decide.
answer options
1. No
2. Yes
From the discriminant of the give quadratic equation, the temperature of the machine will part after 50 minutes of operation.
Will the temperature of the part ever reach or exceed 135°F?The given equation that models the temperature of the machine is;
T = -0.005x² + 0.45x + 125
Let check if there's a value that exists for T = 135
Putting T = 135 in the given equation,
135 = -0.005x² + 0.45x + 125
We can simplify this to;
0.005x² - 0.45x + 10 = 0
From the general form of quadratic equation which is ax² + bx + c = 0, where a = 0.005, b = -0.45, and c = 10.
The discriminant of this quadratic equation is given by:
D = b² - 4ac
= (-0.45)² - 4(0.005)(10)
= 0.2025 - 0.2
= 0.0025
The discriminant of the equation is positive which indicates we have two roots. Therefore, the temperature of the machine part will cross 135°F at some point during the operation.
We can also find the roots of the quadratic equation using the formula:
[tex]x = (-b \± \sqrt(D)) / 2a[/tex]
Substituting the values of a, b, and D, we get:
[tex]x = (0.45 \± \sqrt(0.0025)) / 2(0.005)\\= (0.45 \± 0.05) / 0.01[/tex]
Taking the positive value, we get:
x = 50
Therefore, the temperature of the machine part will cross 135°F after 50 minutes of operation.
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10.5.PS-18 Question content area top Part 1 The diagram shows a track composed of a rectangle with a semicircle on each end. The area of the rectangle is square meters 11200. What is the perimeter of the track? Use 3.14 for pi.
4.1 h(x) Consider h(c) = cos 2x 4.1.1 Complete the table below, rounding your answer off to the first decimal where needed: -90° -75° -60° -45° -30° -15° 0° 15° 30° 45° 60° 75° 90° 4.1.2 Now use the table and draw the graph of h(x) = cos 2x on the system of axes below: -90°-75°-60-45-30-15 2- 14 - 1+ -24 (2) 15° 30° 45° 60⁰ 75⁰ 90⁰ (2) (2)
Here's the completed table and the graph:
x h(x)
-90° 1.0
-75° -0.5
-60° -1.0
-45° -0.0
-30° 1.0
-15° 0.5
0° 1.0
15° 0.5
30° -0.0
45° -1.0
60° -0.5
75° 1.0
90° 1.0
What is function?In mathematics, a function is a relation between a set of inputs and a set of possible outputs, with the property that each input is related to exactly one output. Functions are often represented as a set of ordered pairs, where the first element of each pair is an input and the second element is the corresponding output. Functions are a fundamental concept in many areas of mathematics and have many real-world applications, including in science, engineering, and economics.
Here,
To calculate the values of h(c) in the table, we plug in the given values of x into the function h(c) = cos 2x and evaluate. For example, to find h(c) when x = -75°:
h(c) = cos 2x
h(c) = cos 2(-75°) (substitute -75° for x)
h(c) = cos (-150°) (simplify using the double angle identity)
h(c) = -0.5 (evaluate using the unit circle or a calculator)
We repeat this process for each value of x to fill out the table.
To graph the function h(x) = cos 2x, we plot each point from the table on the given system of axes. The x-axis represents the angle x in degrees, and the y-axis represents the value of h(x) = cos 2x. We then connect the points with a smooth curve to obtain the graph.
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Write these numbers in decreasing order
-4. 1 2/3, 0.5, -1 3/4, 0.03, -1, 1, 0, -103, 54
Answer: 54, 1 2/3, 1, 0.5, 0.03, 0, -1/4, -1, -4, -103
Step-by-step explanation:
54, 1, 1 2/3, 0.5, 0.03, 0, -1/4, -1, -4, -103.
First, we order the numbers by their sign: 54, 1, 1 2/3, 0.5, 0.03, 0, -1/4, -1, -4, -103.
Then we order the positive numbers in decreasing order: 54, 1 2/3, 1, 0.5, 0.03, 0.
Finally, we order the negative numbers in increasing order: -103, -4, -1, -1/4.
Putting it all together, we have: 54, 1 2/3, 1, 0.5, 0.03, 0, -1/4, -1, -4, -103.
The diagrams show three circuits consisting of concentric circular arcs (either half or quarter circles of radii r, 2r, and 3r) and radial lengths. The circuits carry the same current. Rank them according to the magnitudes of the magnetic fields they produce at C, least to greatest
solve correctly and I will pay you $100
The rank of the three circuits consisting of concentric circular arcs according to the magnitudes of the magnetic fields they produce at C, from least to greatest is (3), (2), (1).
We know that, the radial segments don't produce magnetic field at C, so consider arcs.
Assume that the current is counter clockwise and the magnetic field to be positive pointing out of the page.
Understand that, magnetic field at the center from an arc φ of radius R is [tex]\frac{{{\mu _0}i\phi }}{{4\pi R}}[/tex]
Therefore, for (1) :
[tex]\begin{gathered}\begin{array}{l}B = \frac{{{\mu _0}i\pi }}{{4\pi \left( {3r} \right)}} + \frac{{{\mu _0}i\pi }}{{4\pi r}}\\ \Rightarrow B = \frac{1}{3}\frac{{{\mu _0}i}}{r}\end{array}\end{gathered}[/tex]
For (2) :
[tex]\begin{gathered}\begin{array}{l}B = \frac{{{\mu _0}i\pi }}{{4\pi \left( {3r} \right)}} - \frac{{{\mu _0}i\pi }}{{4\pi r}}\\ \Rightarrow B = - \frac{1}{6}\frac{{{\mu _0}i}}{r}\end{array}\end{gathered} \\[/tex]
For (3) :
[tex]\begin{gathered}\begin{array}{l}B = \frac{{{\mu _0}i\pi }}{{4\pi \left( {3r} \right)}} - \frac{{{\mu _0}i\left( {\frac{\pi }{2}} \right)}}{{4\pi r}} - \frac{{{\mu _0}i\left( {\frac{\pi }{2}} \right)}}{{4\pi \left( {2r} \right)}}\\ \Rightarrow B = - \frac{5}{{48}}\frac{{{\mu _0}i}}{r}\end{array}\end{gathered}[/tex]
Therefore, the magnitude of the magnetic fields at C after arranging them in the order of least to greatest are (3), (2), (1).
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Find the value of X using the picture below.
Answer:
x = 7
Step-by-step explanation:
The two angles are equal so the opposite sides are equal.
5x-2 =33
Add two to each side.
5x-2+2 = 33+2
5x=35
Divide by 5
5x/5 =35/5
x = 7
Can someone please help with these 4
Answer:
Step-by-step explanation:
1) b (acute is less than 90)
2) a (obtuse: more than 90, less than 180)
3) c
4) c
Answer:
1. NOM, JOK, KOL
2. MOL, NOK, MOJ
3. NOJ, JOL
4. NOL, MOK
Question 6 of 10
Based only on the information given in the diagram, which congruence
theorems or postulates could be given as reasons why ACDE AOPQ?
Check all that apply.
AA
A. AAS
B. ASA
C. LL
OD. HL
E. LA
F. SAS
Therefore, A, B, C, and F are the proper responses as the congruence theories or postulates based on the data.
what is triangle ?Having three straight sides and three angles where they intersect, a triangle is a closed, two-dimensional shape. It is one of the fundamental geometric shapes and has a number of characteristics that can be used to study and resolve issues that pertain to it. The triangle inequality theory states that the sum of a triangle's interior angles is always 180 degrees, and that the longest side is always the side across from the largest angle. Triangles can be used to solve a wide range of mathematical issues in a variety of disciplines and can be categorised based on the length of their sides and the measurement of their angles.
given
We can use the following congruence theories or postulates based on the data in the diagram:
A. ASA
B. AAS
C. LL (corresponding angles hypothesis)
F. SAS
Therefore, A, B, C, and F are the proper responses as the congruence theories or postulates based on the data.
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Find the absolute maximum and minimum values of the function f(x,y) = x^2+y^2-2x
The function f(x,y) has only minimum value at (1,0) is -1 and maximum value does not exist.
The given function is f(x,y)=x²+y²-2x
First find the partial derivative with respect to x and y
f'(x)=2x-2
f'(y)=2y
f'(x)=0=f'(y)
2x-2=0
x=1
and y=0
Now we will cheak maxima and minima at (1,0)
f''(x,y)=2 and f"(x,y)=2 and f"(x,y)=0( derivative of first order of x with respect to y)
We know that
rt-s²≥0 and r positive then f is minimum and r negative maximum
r=2 , t=2 and s=0
rt-s²≥0 and r is positive so f(x,y) is minimum at (1,0)
f(1,0)=1-2=-1
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Given that x + 1/2 = 5, what is 2*x^2 - 3x + 6 - 3/x +2/x^2
pls help me soon
Sure, let's solve this step-by-step:
First, we need to solve for x in the equation x + 1/2 = 5.
We can do this by subtracting 1/2 from both sides, giving us x = 4 1/2.
Now, we can substitute x = 4 1/2 into the equation 2*x^2 - 3x + 6 - 3/x +2/x^2.
We can simplify the equation by multiplying both sides by x^2, giving us:
2*x^2 - 3x + 6 - 3/x +2 = 10*x^2 - 3x + 6.
Now, we can combine all of the terms with x:
10*x^2 - 6x + 6 = 0.
Finally, we can solve the equation using the quadratic formula:
x = 3/5 or x = 2.
Therefore, the answer to the equation is 10*(3/5)^2 - 6(3/5) + 6 = 4.8, or 10*2^2 - 6(2) + 6 = 16.
Find the value of x. If your answer is not an integer, leave it in simplest radical form. The diagram is not drawn to scale.
NOTE: Enter your answer and show all the steps that you use to solve this problem in the space provided. Use the 30°-60°-90° Triangle Theorem to find the answer.
The value of the x is 5√3 after we successfully do the application of the 30°-60°-90° Triangle theorem.
What is Triangle theorem?The 30°-60°-90° Triangle Theorem states that in such a triangle, the side opposite the 30° angle is half the length of the hypotenuse, and the side opposite the 60° angle is the product of the length of the hypotenuse and the square root of 3 divided by 2.
Using this theorem, we can write:
y = hypotenuse
Opposite of 30° angle = 5 = hypotenuse/2
Opposite of 60° angle = x = hypotenuse × (√(3)/2)
Solving for the hypotenuse in terms of y from the first equation, we get:
hypotenuse = 5×2 = 10
Substituting this value into the third equation, we get:
x = 10 × (√(3)/2) = 5 × √(3)
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Find the local maximum and minimum values of f using both the first and second derivative tests f(x) = x2 / (x - 1). Summary: The local maximum and minimum values of f(x) = x2 / (x - 1) using both the first and second derivative tests is at x = 0 and x = 2.
The value of local maximum and local minimum for the function f(x) = x^2/(x -1 ) is equal to f(0) = 0 at x = 0 and f(2) = 4 at x = 4 respectively.
Local maximum and minimum values of the function
f(x) = x^2 / (x - 1),
Use both the first and second derivative tests.
First, let's find the critical points of the function,
By setting its derivative equal to zero and solving for x,
f'(x) = [2x(x - 1) - x^2] / (x - 1)^2
⇒ [2x(x - 1) - x^2] / (x - 1)^2 = 0
Simplifying this expression, we get,
x(x - 2) = 0
This gives us two critical points,
x = 0 and x = 2.
These critical points correspond to local maxima, local minima, or neither.
Use the second derivative test,
f''(x) = [2(x - 1)^2 - 2x(x - 1) + 2x^2] / (x - 1)^3
At x = 0, we have,
f''(0) = 2 / (-1)^3
= -2
Since the second derivative is negative at x = 0, this critical point corresponds to a local maximum.
f(0) = 0^2/ (0 -1 )
= 0
At x = 2, we have,
f''(2) = 2 / 1^3
= 2
Since the second derivative is positive at x = 2, this critical point corresponds to a local minimum.
f(2) = 2^2/ (2 - 1)
= 4
Therefore, at x = 0, the local maximum value is f(0) = 0, and at x = 2, the local minimum value is f(2) = 4.
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The undergraduate grade point averages (UGPA) of students taking an admissions test in a recent year can be
approximated by a normal distribution, as shown in the figure
(a) What is the minimum UGPA that would still place a student in the top 5% of UGPAS?
(b) Between what two values does the middle 50% of the UGPAS lie?
COLLE
(a) The minimum UGPA that would still place a student in the top 5% of UGPAS is 3.66
(Round to two decimal places as needed.)
(b) The middle 50% of UGPAS lies between 3 26 on the low end and 3.30 on the high end
(Round to two decimal places as needed.)
Between 3.26 on the low end and 3.30 on the high end is where UGPAS's middle 50% lies.
What does a parabola equation mean?Provided that the parabola's vertex is at the origin and that it is symmetric about the y-axis. So, depending on whether the parabola expands upward or downward, the equation can take the form x2 = 4ay or x2 = -4ay.
Because we are interested in the top 5%, the region to the right of the z-score is 0.05. n,... As a result, we can apply the following z-score formula:
z = (x - μ) / σ
x = z * σ + μ
Substituting the values we have, we get:
x = 1.645 * 0.15 + 3.25 = 3.66
Therefore, the z-scores corresponding to the 25th and 75th percentiles are:
z1 = -0.675
z2 = 0.675
Using the same formula as before, we can find the UGPAs corresponding to these z-scares:
x1 = -0.675 * 0.15 + 3.25 = 3.26
x2 = 0.675 * 0.15 + 3.25 = 3.30
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Bria is a customer who would like to display her collection of soap carvings on top of her bookcase. The collection needs an area of 300 square inches. What should b equal for the top of the bookcase to have the correct area? Round your answer to the nearest tenth of an inch. I need help
D: Please !!!!
Answer:
We can use the formula for the area of a rectangle to solve this problem. Let's assume that the length of the top of the bookcase is L and the width is b. Then, we can write:
L × b = 300
Solving for b, we get:
b = 300 / L
Since we don't know the length L, we cannot find the exact value of b. However, we can use the given information to make an estimate. Let's say that the length of the bookcase is 60 inches. Then, we have:
b = 300 / 60 = 5
So, if the length of the bookcase is 60 inches, the width needs to be at least 5 inches to accommodate Bria's soap carving collection. However, if the length is different, the required width will also be different.
Given f(x) = x³ + kx + 9, and the remainder when f(x) is divided by x − 2 is 7,
then what is the value of k?
Answer:
k = -5
Step-by-step explanation:
According to the Remainder Theorem, when we divide a polynomial f(x) by (x − c), the remainder is f(c).
Therefore, if we divide polynomial f(x) = x³ + kx + 9 by (x - 2) and the remainder is 7 then:
f(2) = 7To find the value of k, simply substitute x = 2 into the function, equate it to 7 and solve for k.
[tex]\begin{aligned}f(2)=(2)^3 + k(2) + 9 &= 7\\8+2k+9&=7\\2k+17&=7\\2k&=-10\\k&=-5\end{aligned}[/tex]
Therefore, the value of k is -5.
Locate the absolute extrema of the function on the closed interval
Answer:
The absolute extrema is minimum at (-1, 2/9)
Step-by-step explanation:
Absolute extrema is a logical point that shows whether a the curve function is maximum or minimum.
Forexample a curve in the image attached. A, B and C are points of absolute maxima or absolute maximum. and P and Q are points of absolute minima or minimum.
Remember A, B, C, P, Q are critical points or stationary points.
How do we find absolute extrema?
The find the sign of the second derivative of the function.
From the question;
[tex]{ \sf{g(x) = \sqrt[3]{x} }} \\ \\ { \sf{g(x) = {x}^{ \frac{1}{3} } }} \\ [/tex]
Find the first derivative of g(x)
[tex]{ \sf{g {}^{l}(x) = \frac{1}{3} {x}^{ - \frac{2}{3} } }} \\ [/tex]
Find the second derivative;
[tex]{ \sf{g {}^{ll} (x) = ( \frac{1}{3} \times - \frac{2}{3}) {x}^{( - \frac{2}{3} - 1) } }} \\ \\ { \sf{g {ll}^{(x)} = - \frac{2}{9} {x}^{ - \frac{5}{3} } }}[/tex]
Then substitute for x as -1 from [-1, 1]
[tex]{ \sf{g {}^{ll}(x) = - \frac{2}{9} ( - 1) {}^{ - \frac{5}{3} } }} \\ \\ = \frac{ - 2}{9} \times - 1 \\ \\ = \frac{2}{9} [/tex]
Since the sign of the result is positive, the absolute extrema is minimum
a system of equations is shown below.
y=3x+5
x + y=-7
Answer: x = -3, y = -4, or (-3, -4)
Step-by-step explanation:
To solve the system of equations, we can substitute the first equation into the second equation, replacing y with 3x + 5:
x + (3x + 5) = -7
Simplifying:
4x + 5 = -7
Subtracting 5 from both sides:
4x = -12
Dividing by 4:
x = -3
Now that we know x = -3, we can substitute that value into either of the original equations to find y:
y = 3(-3) + 5 = -4
Therefore, the solution to the system of equations is x = -3, y = -4, or (-3, -4).
7.4y-2.9y
pls lmk....
4.5y
subtract 2.9y from 7.4y, and you get 4.5y
The Venn diagram here shows the cardinality of each set. Use this to find the cardinality of the given set.
n(A)=
The cardinality of set A, n(A) = 29
What is cardinality of a set?The cardinality of a set is the total number of elements in the set
Given the Venn diagram here shows the cardinality of each set. To find the cardinality of set A, n(A), we proceed as follows.
Since the cardinality of a set is the total number of elements in the set, then cardinality of set A , n(A) = 9 + 8 + 3 + 9
= 29
So, n(A) = 29
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PLEASE HELP MEE with all four questionsss
Therefore, the distance between the 90 degree angle and the hypotenuse is approximately 0.829 units.
What is triangle?A triangle is a two-dimensional geometric shape that is formed by three straight line segments that connect to form three angles. It is one of the most basic shapes in geometry and has a wide range of applications in mathematics, science, engineering, and everyday life. Triangles can be classified by the length of their sides (equilateral, isosceles, or scalene) and by the size of their angles (acute, right, or obtuse). The study of triangles is an important part of geometry, and their properties and relationships are used in many areas of mathematics and science.
Here,
1. To find HF, we can use the angle bisector theorem, which states that if a line bisects an angle of a triangle, then it divides the opposite side into two segments that are proportional to the adjacent sides. Let's denote the length of HF as x. Then, by the angle bisector theorem, we have:
JF/FH = JG/HG
Substituting the given values, we get:
15/x = 18/21
Simplifying and solving for x, we get:
x = 15 * 21 / 18
x = 17.5
Therefore, HF is 17.5 cm.
2. Let's denote the length of the hypotenuse as h and the length of the leg opposite the 18-unit perpendicular as a. We can then use the Pythagorean theorem to write:
h² = a² + 18²
We are told that the hypotenuse is divided into segments of length x and 6 units, so we can write:
h = x + 6
Substituting this expression into the first equation, we get:
(x + 6)² = a² + 18²
We are also told that the leg adjacent to the angle opposite the 4-unit segment is divided into segments of length 4 and (a - 4), so we can write:
a = 4 + (a - 4)
Simplifying this equation, we get:
a = a
Now we can substitute this expression for a into the previous equation and solve for x:
(x + 6)² = (4 + (a - 4))² + 18²
Expanding and simplifying, we get:
x² + 12x - 36 = 0
Using the quadratic formula, we get:
x = (-12 ± √(12² - 4(1)(-36))) / (2(1))
x = (-12 ± √(288)) / 2
x = -6 ± 6√(2)
Since the length of a segment cannot be negative, we take the positive root:
x = -6 + 6sqrt(2)
x ≈ 1.46
Therefore, the value of x is approximately 1.46 units.
3. Let's denote the length of the hypotenuse as h and the length of the leg adjacent to the angle opposite the 9-unit perpendicular as b. We can then use the Pythagorean theorem to write:
h² = b² + 9²
We are told that the hypotenuse is divided into segments of length x and 6 units, so we can write:
h = x + 6
Substituting this expression into the first equation, we get:
(x + 6)² = b² + 9²
Expanding and simplifying, we get:
x² + 12x - b² = 27
We also know that the length of the leg opposite the 9-unit perpendicular is:
a = √(h² - 9²)
= √((x + 6)² - 9²)
= √(x² + 12x + 27)
Now we can use the fact that the tangent of the angle opposite the 9-unit perpendicular is equal to the ratio of the lengths of the opposite and adjacent sides:
tan(θ) = a / b
Substituting the expressions for a and b, we get:
tan(θ) = √(x² + 12x + 27) / (x + 6)
We also know that the tangent of the angle theta is equal to the ratio of the length of the opposite side to the length of the adjacent side:
tan(θ) = 9 / b
Substituting the expression for b, we get:
tan(θ) = 9 / √(h² - 9²)
Substituting the expression for h, we get:
tan(θ) = 9 / √((x + 6)² - 9²)
Since the tangent function is the same for equal angles, we can set these two expressions for the tangent equal to each other:
√(x² + 12x + 27) / (x + 6) = 9 / √((x + 6)² - 9²)
Squaring both sides, we get:
(x² + 12x + 27) / (x + 6)² = 81 / ((x + 6)² - 81)
Cross-multiplying and simplifying, we get:
x⁴ + 36x³ + 297x² - 1458x - 2916 = 0
Using a numerical method such as the Newton-Raphson method or the bisection method, we can find the approximate solution to this equation:
x ≈ 9.449
Therefore, the value of x is approximately 9.449 units.
4. Let's denote the length of the hypotenuse as h and the length of the leg adjacent to the angle opposite the distance we want to find as b. We can use the Pythagorean theorem to write:
h² = b² + d²
We are told that the hypotenuse is divided into segments of length 9 and 4 units, so we can write:
h = 9 + 4 = 13
Substituting this expression into the first equation, we get:
13² = b² + d²
Simplifying and solving for d, we get:
d = √(13² - b²)
Now, we need to find the value of b. We know that the hypotenuse is divided into segments of length 9 and 4 units, so we can use similar triangles to write:
b / 4 = 9 / 13
Simplifying and solving for b, we get:
b = 36 / 13
Substituting this expression for b into the equation we found earlier for d, we get:
d = √(13² - (36/13)²)
Simplifying and finding a common denominator, we get:
d =√ (169*13 - 36²) / 13²
Simplifying further, we get:
d = √(169169 - 3636) / 169
Calculating this expression, we get:
d ≈ 0.829
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One reason for using a distribution instead of the standard Normal curve to find critical values when calculating a level C confidence interval for a population mean is that
(a) z can be used only for large samples.
(b) z requires that you know the population standard deviation θ
.
(c) z requires that you can regard your data as an SRS from the population.
(d) the standard Normal table doesn't include confidence levels at the bottom.
(e) a z critical value will lead to a wider interval than a t critical value.
(b) z requires that you know the population standard deviation θ
.
Therefore , the solution of the given problem of standard deviation comes out to be the group standard deviation in order to use (b) z.
What does standard deviation actually mean?Statistics uses variance as a way to quantify difference. The image of the result is used to compute the average deviation between the collected data and the mean. Contrary to many other valid measures of variability, it includes those pieces of data on their own by comparing each number to the mean. Variations may be caused by willful mistakes, irrational expectations, or shifting economic or business conditions.
Here,
We use the z-distribution if the total standard deviation is known; otherwise, we use the t-distribution.
Additionally, for small sample sizes, the t-distribution is used, whereas for big sample sizes, the z-distribution is used.
The fact that z requires that you know the population standard deviation, and that this is frequently not known in practice, is one reason to use a distribution rather than the traditional .
Normal curve to find critical values when computing a level C confidence interval for a population mean.
You must be aware of the group standard deviation in order to use (b) z.
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CALCULUS HELP NEEDED: Express the integrand as a sum of partial fractions and evaluate the integrals.
[tex]\int\ {\frac{x+3}{2x^{3}-8x}} \, dx[/tex]
**I know I need to solve for A&B, but I have no idea where to start for partial fractions.
The integral of the function expressed as sum of partial frictions is -3/8 ln|x| + 7/8 ln|x + 2| - 1/8 ln|x - 2| + C.
What is the integral of function?
First, factor out 2x from the denominator to obtain:
∫[(x + 3)/(2x³ - 8x)] dx = ∫[(x + 3)/(2x)(x² - 4)] dx
Next, we use partial fractions to express the integrand as a sum of simpler fractions. To do this, we need to factor the denominator of the integrand:
2x(x² - 4) = 2x(x + 2)(x - 2)
Therefore, we can write:
(x + 3)/(2x)(x² - 4) = A/(2x) + B/(x + 2) + C/(x - 2)
Multiplying both sides by the denominator, we get:
x + 3 = A(x + 2)(x - 2) + B(2x)(x - 2) + C(2x)(x + 2)
Now, we need to find the values of A, B, and C. We can do this by equating coefficients of like terms:
x = A(x² - 4) + B(2x² - 4x) + C(2x² + 4x)
x = (A + 2B + 2C)x² + (-4A - 4B + 4C)x - 4A
Equating coefficients of x², x, and the constant term, respectively, we get:
A + 2B + 2C = 0
-4A - 4B + 4C = 1
-4A = 3
Solving for A, B, and C, we find:
A = -3/4
B = 7/16
C = -1/16
Therefore, the partial fraction decomposition is:
(x + 3)/(2x)(x² - 4) = -3/(4(2x)) + 7/(16(x + 2)) - 1/(16(x - 2))
The integral becomes:
∫[(x + 3)/(2x³ - 8x)] dx = ∫[-3/(8x) + 7/(8(x + 2)) - 1/(8(x - 2))] dx
Integrating each term separately gives:
∫[-3/(8x) + 7/(8(x + 2)) - 1/(8(x - 2))] dx
= -3/8 ln|x| + 7/8 ln|x + 2| - 1/8 ln|x - 2| + C
where;
C is the constant of integration.Therefore, the final answer is:
∫[(x + 3)/(2x³ - 8x)] dx = -3/8 ln|x| + 7/8 ln|x + 2| - 1/8 ln|x - 2| + C
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There is 32 students in class.8 boys out of 20 wear cap and 8 girls out of 12 wear spectacles find the probability that the student is boy who wears cap or girls who doesn't wear spectacles
Answer:
Boy = 40%
Girl = 66.6%
Step-by-step explanation:
1) Work out probability that student is a boy who wears cap
8 boys out of 20 wears a cap so to find the probability we have to do 8 divided by 20
8 ÷ 20 = 0.40.4 × 100 = 40%2) Work out probability for girls who doesn't wear spectacles
To find the probability of girls who doesn't wear spectacles we have to do 8 divided by 12
8 ÷ 12 = 0.666...(recurring)0.666...(recurring) × 100 = 66.6%Hope this helps, have a lovely day! :)
help I’ll give brainliest ^•^ just question (7) thanks!!
Answer:
To shift the graph of f(x) = |x| to have a domain of [-3, 6], we need to move the left endpoint from -6 to -3 and the right endpoint from 3 to 6.
A translation to the right by 3 units will move the left endpoint of the graph of f(x) to -3, but it will also shift the right endpoint to 6 + 3 = 9, which is outside the desired domain.
A translation to the left by 3 units will move the right endpoint of the graph of f(x) to 3 - 3 = 0, which is outside the desired domain.
A translation upward or downward will not change the domain of the graph, so options B and D can be eliminated.
Therefore, the correct answer is C g(x) = x - 3. This translation will move the left endpoint to -3 and the right endpoint to 6, which is exactly the desired domain.
According to Money magazine, Maryland had the highest median annual household income of any state in 2018 at $75,847.† Assume that annual household income in Maryland follows a normal distribution with a median of $75,847 and standard deviation of $33,800.
(a) What is the probability that a household in Maryland has an annual income of $90,000 or more? (Round your answer to four decimal places.)
(b) What is the probability that a household in Maryland has an annual income of $50,000 or less? (Round your answer to four decimal places.)
The required probability that a household in Maryland with annual income of ,
$90,000 or more is equal to 0.3377.
$50,000 or less is equal to 0.2218.
Annual household income in Maryland follows a normal distribution ,
Median = $75,847
Standard deviation = $33,800
Probability of household in Maryland has an annual income of $90,000 or more.
Let X be the random variable representing the annual household income in Maryland.
Then,
find P(X ≥ $90,000).
Standardize the variable X using the formula,
Z = (X - μ) / σ
where μ is the mean (or median, in this case)
And σ is the standard deviation.
Substituting the given values, we get,
Z = (90,000 - 75,847) / 33,800
⇒ Z = 0.4187
Using a standard normal distribution table
greater than 0.4187 as 0.3377.
P(X ≥ $90,000)
= P(Z ≥ 0.4187)
= 0.3377
Probability that a household in Maryland has an annual income of $90,000 or more is 0.3377(rounded to four decimal places).
Probability that a household in Maryland has an annual income of $50,000 or less.
P(X ≤ $50,000).
Standardizing X, we get,
Z = (50,000 - 75,847) / 33,800
⇒ Z = -0.7674
Using a standard normal distribution table
Probability that a standard normal variable is less than -0.7674 as 0.2218. This implies,
P(X ≤ $50,000)
= P(Z ≤ -0.7674)
= 0.2218
Probability that a household in Maryland has an annual income of $50,000 or less is 0.2218.
Therefore, the probability with annual income of $90,000 or more and $50,000 or less is equal to 0.3377 and 0.2218 respectively.
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Answer this imagine please
The expression that is not equivalent to the model shown is given as follows:
-4(3 + 2). -> Option C.
What are equivalent expressions?Equivalent expressions are mathematical expressions that have the same value, even though they may look different. In other words, two expressions are equivalent if they produce the same output for any input value.
The expression for this problem is given by three times the subtraction of four, plus three times the addition of 2, hence:
3(-4) + 3(2) = -12 + 6 = 3(-4 + 2) = 3(-2) = -6.
Hence the expression that is not equivalent is the expression given in option C, for which the result is given as follows:
-4(3 + 2) = -4 x 5 = -20.
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