Answer:
Letters can be chosen in 12 different ways, if order matters, or 6 different ways, if order doesn't matter.
Step-by-step explanation:
Since we want to choose 2 letters, without replacement, from the 4 letters A, B, C, and D, to determine in how many ways can this be done, if the order of the choices matters, and in how many ways can this be done, if the order of the choices does not matter, the following calculations must be performed:
If order matters =
(4 x 3 x 2 x 1) / 2 = X
24/2 = X
12 = X
If the order doesn't matter =
12/2 = X
6 = X
Therefore, letters can be chosen in 12 different ways, if order matters, or 6 different ways, if order doesn't matter.
A group of rowdy teenagers near a wind turbine decide to place a pair of pink shorts on the tip of one blade, they notice the shorts are at its maximum height of 16 meters at a and it’s minimum height of 2 meters at s.
Determine the equation of the sinusoidal function that describes the height of the shorts in terms of time.
Determine the height of the shorts exactly t=10 minutes, to the nearest tenth of a meter
The equation of the sinusoidal function is 7 × sin((π/15)·(x - 2.5)) + 9
Question: The likely missing parameters in the question are;
The time at which the shorts are at the maximum height, t₁ = 10 seconds
The time at which the shorts are at the minimum height, t₂ = 25 seconds
The general form of a sinusoidal function is A·sin(B(x - h)) + kWhere;
A = The amplitude
The period, T = 2·π/B
The horizontal shift = h
The vertical shift = k
The parent equation of the sine function = sin(x)
We find the values of the variables, A, B, h, and k as follows;
The given parameters of the sinusoidal function are;
The maximum height = 16 meters at time t₁ = 10 seconds
The minimum height = 2 meters at time t₂ = 25 seconds
The time it takes the shorts to complete a cycle, (maximum height to maximum height), the period, T = 2 × (t₂ - t₁)
∴ T = 2 × (25 - 10) = 30
The amplitude, A = (Maximum height- Minimum height)/2
∴ A = (16 m - 2 m)/2 = 7 m
The amplitude of the motion, A = 7 meters
T = 2·π/B
∴ B = 2·π/T
T = 30 seconds
∴ B = 2·π/30 = π/15
B = π/15
At t = 10, y = Maximum
Therefore;
sin(B(x - h)) = Maximum, which gives; (B(x - h)) = π/2
Plugging in B = π/15, and t = 10, gives;
((π/15)·(10 - h)) = π/2
10 - h = (π/2) × (15/π) = 7.5
h = 10 - 7.5 = 2.5
h = 2.5
The minimum value of a sinusoidal function, having a centerline of which is on the x-axis, and which has an amplitude, A, is -A
Therefore, the minimum value of the motion of the turbine blades before, the vertical shift = -A = -7
The given minimum value = 2
The vertical shift, k = 2 - (-7) = 9
Therefore, k = 9
Therefore;
The equation of the sinusoidal function is 7 × sin((π/15)·(x - 2.5)) + 9
More can be learned about sinusoidal functions on Brainly here;
https://brainly.com/question/14850029
Solve x^2 - 8x - 9 = 0.
Rewrite the equation so that it is of the form
x2 + bx = c.
x^2+__x=__
Answer:
x^2+8x =9 equal to x^2+8x-9=0
(x+9)(x-1)
Step-by-step explanation:
x+9=0
x=-9
OR
x-1=0
x=+1
The equation can be rewritten as x² + (-8)x = 9.
What are Quadratic Equations?Quadratic equations are polynomial equations of second degree.
The general form of a quadratic equation is ax² + b x + c = 0.
Given quadratic equation is,
x² - 8x - 9 = 0
We have to rewrite in the form,
x2 + bx = c.
Adding 9 to both sides of the equation,
x² - 8x - 9 + 9 = 0 + 9
x² - 8x = 9
x² + (-8)x = 9
Hence the rewritten form of the equation is x² + (-8)x = 9.
Learn more about Quadratic Equations here :
https://brainly.com/question/30098550
#SPJ7
Solve the equation by completing the square.
0 = 4x2 − 72x
Answer:
B
Step-by-step explanation:
Given
4x² - 72x = 0 ← factor out 4 from each term
4(x² - 18x) = 0
To complete the square
add/subtract (half the coefficient of the x- term)² to x² - 18x
4(x² + 2(- 9)x + 81 - 81) = 0
4(x - 9)² - 4(81) = 0
4(x - 9)² - 324 = 0 ( add 324 to both sides )
4(x - 9)² = 324 ( divide both sides by 4 )
(x - 9)² = 81 ( take the square root of both sides )
x - 9 = ± [tex]\sqrt{81}[/tex] = ± 9 ( add 9 to both sides )
x = 9 ± 9
Then
x = 9 - 9 = 0
x = 9 + 9 = 18
Answer:0,18
Step-by-step explanation:
its right
Graph the inequality.
7 <= y - 2x < 12
Answer:
X(-12,-7)
Step-by-step explanation:
This is the answer to your problem. I hope it helps. I don't know how to explain it sorry.
Which of the following exponential equations is equivalent to the logarithmic
equation below?
log 970 = x
A.x^10-970
B. 10^x- 970
C. 970^x- 10
D. 970^10- X
Given:
The logarithmic equation is:
[tex]\log 970=x[/tex]
To find:
The exponential equations that is equivalent to the given logarithmic equation.
Solution:
Property of logarithm:
If [tex]\log_b a=x[/tex], then [tex]a=b^x[/tex]
We know that the base log is always 10 if it is not mentioned.
If [tex]\log a=x[/tex], then [tex]a=10^x[/tex]
We have,
[tex]\log 970=x[/tex]
Here, base is 10 and the value of a is 970. By using the properties of exponents, we get
[tex]970=10^x[/tex]
Interchange the sides, we get
[tex]10^x=970[/tex]
Therefore, the correct option is B, i.e., [tex]10^x=970[/tex].
Note: It should be "=" instead of "-" in option B.
According to the National Association of Theater Owners, the average price for a movie in the United States in 2012 was $7.96. Assume the population st. dev. is $0.50 and that a sample of 30 theaters was randomly selected. What is the probability that the sample mean will be between $7.75 and $8.20
Answer:
0.985 = 98.5% probability that the sample mean will be between $7.75 and $8.20.
Step-by-step explanation:
To solve this question, we need to understand the normal probability distribution and the central limit theorem.
Normal Probability Distribution
Problems of normal distributions can be solved using the z-score formula.
In a set with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the z-score of a measure X is given by:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the p-value, we get the probability that the value of the measure is greater than X.
Central Limit Theorem
The Central Limit Theorem establishes that, for a normally distributed random variable X, with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean [tex]\mu[/tex] and standard deviation [tex]s = \frac{\sigma}{\sqrt{n}}[/tex].
For a skewed variable, the Central Limit Theorem can also be applied, as long as n is at least 30.
The average price for a movie in the United States in 2012 was $7.96. Assume the population st. dev. is $0.50.
This means that [tex]\mu = 7.96, \sigma = 0.5[/tex]
Sample of 30:
This means that [tex]n = 30, s = \frac{0.5}{\sqrt{30}}[/tex]
What is the probability that the sample mean will be between $7.75 and $8.20?
This is the p-value of Z when X = 8.2 subtracted by the p-value of Z when X = 7.75.
X = 8.2
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
By the Central Limit Theorem
[tex]Z = \frac{X - \mu}{s}[/tex]
[tex]Z = \frac{8.2 - 7.96}{\frac{0.5}{\sqrt{30}}}[/tex]
[tex]Z = 2.63[/tex]
[tex]Z = 2.63[/tex] has a p-value of 0.9957
X = 7.75
[tex]Z = \frac{X - \mu}{s}[/tex]
[tex]Z = \frac{7.75 - 7.96}{\frac{0.5}{\sqrt{30}}}[/tex]
[tex]Z = -2.3[/tex]
[tex]Z = -2.3[/tex] has a p-value of 0.0107.
0.9957 - 0.0157 = 0.985
0.985 = 98.5% probability that the sample mean will be between $7.75 and $8.20.
!!!!Please Answer Please!!!!
ASAP!!!!!!
!!!!!!!!!!!!!
Answer:
False
Step-by-step explanation:
well i think that the answer from my calculations
What is the measure of F?
G
65
10
H H
10
A. Cannot be determined
B. 55
C. 75
D65
Answer:
D. 65°
Step-by-step explanation:
It is so because the triangle is isosceles, two identical sides and two equal angles.
Find the interval in which y= x2 + 4 is increasing
Answer:
x > 0 and x > -6
[tex]{ \tt{y = {x}^{2} + 4}} \\ { \tt{for \: x > 0 :positive \: integers }} \\ y = 0 < x < {}^{ + } \infin[/tex]
An isosceles right triangle has a hypotenuse that measures 4√2 cm. What is the area of the triangle?
PLEASE HELP
Answer:
8
Step-by-step explanation:
As it's an isosceles right triangle, it's sides are equal, say x. x^2+x^2=(4*sqrt(2))^2. x=4, Area is (4*4)/2=8
The polynomial equation x cubed + x squared = negative 9 x minus 9 has complex roots plus-or-minus 3 i. What is the other root? Use a graphing calculator and a system of equations. –9 –1 0 1
9514 1404 393
Answer:
(b) -1
Step-by-step explanation:
The graph shows the difference between the two expressions is zero at x=-1.
__
Additional comment
For finding solutions to polynomial equations, I like to put them in the form f(x)=0. Most graphing calculators find zeros (x-intercepts) easily. Sometimes they don't do so well with points where curves intersect. Also, the function f(x) is easily iterated by most graphing calculators in those situations where the root is irrational or needs to be found to best possible accuracy.
Answer:
The answer is b: -1
Step-by-step explanation:
good luck!
I dont get what this is asking me to do
Answer:
Step-by-step explanation:
what this statement is saying is that if you have
4 x + 3x^2 + 5 + 8x^2 + 12x + 9
that
3x^2 + 8x^2 = 11x^2
4 x + 12x = 16x
5 + 9 = 14
get added together the final answer would be
in descending order as : 11x^2 + 16x + 14
Pleaseee Help. What is the value of x in this simplified expression?
(-1) =
(-j)*
1
X
What is the value of y in this simplified expression?
1 1
ky
y =
-10
K+m
+
.10
m т
9514 1404 393
Answer:
x = 7
y = 5
Step-by-step explanation:
The applicable rule of exponents is ...
a^-b = 1/a^b
__
For a=-j and b=7,
(-j)^-7 = 1/(-j)^7 ⇒ x = 7
For a=k and b=-5,
k^-5 = 1/k^5 ⇒ y = 5
While out for a run, two joggers with an average age of 55 are joined by a group of three more joggers with an average age of m. if the average age of the group of five joggers is 45, which of the following must be true about the average age of the group of 3 joggers?
a) m=31
b) m>43
c) m<31
d) 31 < m < 43
Answer:
they have it on calculator soup
Step-by-step explanation:
Answer:
D. 31<m<43
Step-by-step explanation:
45 x 5 = 225 which is the age of the 5 joggers altogether.
55 x 2 = 110 which is the age of the 2 joggers together.
3m + 110 = 225 then solve for m so,
3m = 115
m = 38.3333
so hence, m is greater than 31 but less than 43.
answer: D
(4-1) + (6 + 5) = help plz
What is the value of x in the triangle? 45, 45, x
Answer:
90
Step-by-step explanation:
it its a 45 45 90 triangle
the point (-2,5) is reflected across the y-axis. which of these is the ordered pair of the image
Answer:(2,5)
Step-by-step explanation: watch this video
https://youtu.be/l78P2Xi68-k
if 3 sec²θ-5tan θ-4=0 find the general solution to this equation
3 sec²(θ) - 5 tan(θ) - 4 = 0
Recall the Pythagorean identity,
cos²(θ) + sin²(θ) = 1.
Multiplying both sides by 1/cos²(θ) gives another form of the identity,
1 + tan²(θ) = sec²(θ).
Then the equation becomes quadratic in tan(θ):
3 (1 + tan²(θ)) - 5 tan(θ) - 4 = 0
3 tan²(θ) - 5 tan(θ) - 1 = 0
I'll solve by completing the square.
tan²(θ) - 5/3 tan(θ)) - 1/3 = 0
tan²(θ) - 5/3 tan(θ) = 1/3
tan²(θ) - 5/3 tan(θ) + 25/36 = 1/3 + 25/36
(tan(θ) - 5/6)² = 37/36
tan(θ) - 5/6 = ±√37/6
tan(θ) = (5 ± √37)/6
Take the inverse tangent of both sides:
θ = arctan((5 + √37)/6) + nπ or θ = arctan((5 - √37)/6) + nπ
where n is any integer
Pleas help me in this question Find R
Answer:
R = 25.8
Step-by-step explanation:
Since this is a right triangle, we can use trig functions
cos R = adj side / hyp
cos R = 9/10
Taking the inverse cos of each side
cos ^-1 ( cos R) = cos^ -1 ( 9/10)
R=25.84193
Rounding to the nearest tenth
R = 25.8
Answer:
[tex]\boxed {\boxed {\sf D. \ 25.8 \textdegree} }[/tex]
Step-by-step explanation:
We are asked to find the measure of an angle given the triangle with 2 sides. This is a right triangle because of the small square representing a right angle. Therefore, we can use trigonometric functions. The three major functions are:
sinθ= opposite/hypotenuse cosθ= adjacent/hypotenuse tanθ= opposite/adjacentWe are solving for angle R, and we have the sides TR (measures 9) and SR (measures 10).
The side TR (9) is adjacent or next to angle R. The side SR (10) is the hypotenuse because it is opposite the right angle.We have the adjacent side and the hypotenuse, so we will use the cosine function.
[tex]cos \theta = \frac {adjacent}{hypotenuse}[/tex]
[tex]cos R = \frac {9}{10}[/tex]
Since we are solving for an angle, we must take the inverse cosine of both sides.
[tex]cos^{-1}(cos R) = cos ^{-1} ( \frac{9}{10})[/tex]
[tex]R = cos ^{-1} ( \frac{9}{10})[/tex]
[tex]R= 25.84193276[/tex]
If we round to the nearest tenth, the 4 in the hundredth place tells us to leave the 8 in the tenths place.
[tex]R \approx 25.8 \textdegree[/tex]
The measure of angle R is approximately 25.8 degrees and choice D is correct.
What is the complete factorization of the polynomial below?
x3 + 8x2 + 17x + 10
A. (x + 1)(x + 2)(x + 5)
B. (x + 1)(x-2)(x-5)
C. (x-1)(x+2)(x-5)
O D. (x-1)(x-2)(x + 5)
Answer: A (x+1)(x+2)(x+5)
Step-by-step explanation:
Below are the heights (in inches) of students in a third-grade class. Find the median height. 39, 37, 48, 49, 40, 42, 48, 53, 47, 42, 49, 51, 52, 45, 47, 48
Given:
The heights (in inches) of students in a third-grade class are:
39, 37, 48, 49, 40, 42, 48, 53, 47, 42, 49, 51, 52, 45, 47, 48
To find:
The median height.
Solution:
The given data set is:
39, 37, 48, 49, 40, 42, 48, 53, 47, 42, 49, 51, 52, 45, 47, 48
Arrange the data set in ascending order.
37, 39, 40, 42, 42, 45, 47, 47, 48, 48, 48, 49, 49, 51, 52, 53
Here, the number of observations is 16. So, the median of the given data set is:
[tex]Median=\dfrac{\dfrac{n}{2}\text{th term}+\left(\dfrac{n}{2}+1\right)\text{th term}}{2}[/tex]
[tex]Median=\dfrac{\dfrac{16}{2}\text{th term}+\left(\dfrac{16}{2}+1\right)\text{th term}}{2}[/tex]
[tex]Median=\dfrac{8\text{th term}+9\text{th term}}{2}[/tex]
[tex]Median=\dfrac{47+48}{2}[/tex]
[tex]Median=\dfrac{95}{2}[/tex]
[tex]Median=47.5[/tex]
Therefore, the median height of the students is 47.5 inches.
When Cameron moved into a new house, he planted two trees in his backyard. At the time of planting, Tree A was 24 inches tall and Tree B was 39 inches tall. Each year thereafter, Tree A grew by 6 inches per year and Tree B grew by 3 inches per year. Let AA represent the height of Tree A tt years after being planted and let BB represent the height of Tree B tt years after being planted. Write an equation for each situation, in terms of t,t, and determine the interval of time, t,t, when Tree A is taller than Tree B.
Answer:
time interval when Tree A is taller than Tree B is;
t > 5 years
Step-by-step explanation:
Tree A;
Initial height = 24 inches
Increase in height per year = 6 inches per year
Thus, for t years after being planted, height is;
A = 6t + 24
Tree B;
Initial height = 39 inches
Increase in height per year = 3 inches per year
Thus, for t years after being planted, height is;
B = 3t + 39
For tree A to be taller than tree B, then it means thay;
A > B
Thus;
6t + 24 > 3t + 39
Subtract 3t from both sides to get;
6t - 3t + 24 > 39
3t + 24 > 39
3t > 39 - 24
3t > 15
Divide both sides by 3 to get;
t > 5
Thus, time interval when Tree A is taller than Tree B is; t > 5
Prove the following identities : i) tan a + cot a = cosec a sec a
Step-by-step explanation:
[tex]\tan \alpha + \cot\alpha = \dfrac{\sin \alpha}{\cos \alpha} +\dfrac{\cos \alpha}{\sin \alpha}[/tex]
[tex]=\dfrac{\sin^2\alpha + \cos^2\alpha}{\sin\alpha\cos\alpha}=\dfrac{1}{\sin\alpha\cos\alpha}[/tex]
[tex]=\left(\dfrac{1}{\sin\alpha}\right)\!\left(\dfrac{1}{\cos\alpha}\right)=\csc \alpha \sec\alpha[/tex]
Question :
tan alpha + cot Alpha = cosec alpha. sec alphaRequired solution :
Here we would be considering L.H.S. and solving.
Identities as we know that,
[tex] \red{\boxed{\sf{tan \: \alpha \: = \: \dfrac{sin \: \alpha }{cos \: \alpha} }}}[/tex][tex] \red{\boxed{\sf{cot \: \alpha \: = \: \dfrac{cos \: \alpha }{sin \: \alpha} }}}[/tex]By using the identities we gets,
[tex] : \: \implies \: \sf{ \dfrac{sin \: \alpha }{cos \: \alpha} \: + \: \dfrac{cos \: \alpha }{sin \: \alpha} }[/tex]
[tex]: \: \implies \: \sf{ \dfrac{sin \: \alpha \times sin \: \alpha }{cos \: \alpha \times sin \: \alpha} \: + \: \dfrac{cos \: \alpha \times cos \: \alpha }{sin \: \alpha \times \: cos \: \alpha } } [/tex]
[tex] : \: \implies \: \sf{ \dfrac{sin {}^{2} \: \alpha }{cos \: \alpha \times sin \alpha} \: + \: \dfrac{cos {}^{2} \: \alpha }{sin \: \alpha \times \: cos \: \alpha } } [/tex]
[tex]: \: \implies \: \sf{ \dfrac{sin {}^{2} \: \alpha }{cos \: \alpha \: sin \alpha} \: + \: \dfrac{cos {}^{2} \: \alpha }{sin \: \alpha \: cos \: \alpha } } [/tex]
[tex]: \: \implies \: \sf{ \dfrac{sin {}^{2} \: \alpha \: + \: cos {}^{2} \alpha}{cos \: \alpha \: sin \alpha} } [/tex]
Now, here we would be using the identity of square relations.
[tex]\red{\boxed{ \sf{sin {}^{2} \alpha \: + \: cos {}^{2} \alpha \: = \: 1}}}[/tex]By using the identity we gets,
[tex] : \: \implies \: \sf{ \dfrac{1}{cos \: \alpha \: sin \alpha} }[/tex]
[tex]: \: \implies \: \sf{ \dfrac{1}{cos \: \alpha } \: + \: \dfrac{1}{sin\: \alpha} }[/tex]
[tex]: \: \implies \: \bf{sec \alpha \: cosec \: \alpha}[/tex]
Hence proved..!!Assume the readings on thermometers are normally distributed with a mean of 0°C and a standard deviation of 1.00°C. Find the probability that a randomly selected thermometer reads between −2.23 and −1.69 and draw a sketch of the region.
Answer:
0.0326 = 3.26% probability that a randomly selected thermometer reads between −2.23 and −1.69.
The sketch is drawn at the end.
Step-by-step explanation:
Normal Probability Distribution
Problems of normal distributions can be solved using the z-score formula.
In a set with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the z-score of a measure X is given by:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the p-value, we get the probability that the value of the measure is greater than X.
Mean of 0°C and a standard deviation of 1.00°C.
This means that [tex]\mu = 0, \sigma = 1[/tex]
Find the probability that a randomly selected thermometer reads between −2.23 and −1.69
This is the p-value of Z when X = -1.69 subtracted by the p-value of Z when X = -2.23.
X = -1.69
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{-1.69 - 0}{1}[/tex]
[tex]Z = -1.69[/tex]
[tex]Z = -1.69[/tex] has a p-value of 0.0455
X = -2.23
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{-2.23 - 0}{1}[/tex]
[tex]Z = -2.23[/tex]
[tex]Z = -2.23[/tex] has a p-value of 0.0129
0.0455 - 0.0129 = 0.0326
0.0326 = 3.26% probability that a randomly selected thermometer reads between −2.23 and −1.69.
Sketch:
Which of the following is the differnce of two squares
what is the solution to the equation?
Answer:
Step-by-step explanation:
log(20x³) - 2logx = 4
log(20x³) -log(x²) = 4
log(20x³/x²) = 4
log(20x) = 4
20x = 10⁴
x = 10⁴/20 = 500
X+34>55
Solve the inequality and enter your solution as an inequality comparing the variable to a number
Answer:
x > 21
General Formulas and Concepts:
Pre-Algebra
Equality Properties
Multiplication Property of Equality Division Property of Equality Addition Property of Equality Subtraction Property of EqualityStep-by-step explanation:
Step 1: Define
Identify
x + 34 > 55
Step 2: Solve for x
[Subtraction Property of Equality] Subtract 34 on both sides: x > 21An equation is shown below:
3(4x − 2) = 1
Which of the following correctly shows the steps to solve this equation?
Step 1: 12x − 2 = 1; Step 2: 12x = 3
Step 1: 12x − 6 = 1; Step 2: 12x = 7
Step 1: 7x + 1 = 1; Step 2: 7x = 0
Step 1: 7x − 5 = 1; Step 2: 7x = 6
Step-by-step explanation:
Step 1: 12x-6= 1
step 2:12x=7
lcm of fractions 7/12, 3/8, and 11/36
A homeowner estimates that it will take 11 days to roof his house. A professional roofer estimates that he could roof the house in 6 days. How long will it take if the homeowner helps the roofer?
Answer:
If the homeowner helps the roofer it will take 4 days to roof the house.
Step-by-step explanation:
Given that a homeowner estimates that it will take 11 days to roof his house, while a professional roofer estimates that he could roof the house in 6 days, to determine how long it will take if the homeowner helps the roofer the following calculation must be done:
1/6 + 1/11 = X
0.1666666 + 0.09090909 = X
0.25 = X
1/4 = X
Therefore, if the homeowner helps the roofer it will take 4 days to roof the house.
