Answer:
125
Step-by-step explanation:
We Take
625 / 5 = 125
So, the answer is 125
Point E represents the center of this circle. Angle DEF
has a measure of 80%.
Drag and drop a number into the box to correctly
complete the statement.
An angle measure of 80° is the size of an angle
that turns through
20
50
one-degree turns.
80
100
K
The measure of the arc intercepted by the angle and the vertical angles make up the angle subtended at the center. As a result, XYZ has a value of 35°.
What are angles?Two lines intersect at a location, creating an angle.
An "angle" is the term used to describe the width of the "opening" between these two rays. The character is used to represent it.
Angles are frequently expressed in degrees and radians, a unit of circularity or rotation.
In geometry, an angle is created by joining two rays at their ends. These rays are referred to as the angle's sides or arms.
An angle has two primary components: the arms and the vertex. T
he two rays' shared vertex serves as their common terminal.
Hence, The measure of the arc intercepted by the angle and the vertical angles make up the angle subtended at the center. As a result, XYZ has a value of 35°.
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Oliver's normal rate of pay is $10.40 an hour.
How much is he paid for working 5 hours overtime one Saturday at time-and-a-half?
12. If zo 125°, what does zz equal in this figure?
A. 125°
B. 180°
C. 35°
D. 55°
Answer:
A
Step-by-step explanation:
∠ o and ∠ z are alternate exterior angles and are congruent, that is
∠ z = ∠ o = 125°
A company rents storage sheds shaped like rectangular prisms. Each shed is 11 feet long, 7 feet wide, and 12 feet tall. The rental cost is $3 per cubic foot. How much does it cost to rent one shed?
The cost to rent one shed of the rectangular prism shaped shed is $2772.
What is area?The size of a section on a surface is determined by its area. Surface area refers to the area of an open surface or the border of a three-dimensional object, whereas the area of a plane region or plane area refers to the area of a shape or planar lamina.
What is a prism?A rectangular prism is a polyhedron in geometry that has two parallel and congruent sides. It also goes by the name cuboid. Six faces, each with a rectangle form and twelve edges, make up a rectangular prism. It is referred to as a prism because of the extent of its cross-section.
Volume of prism= BH
where B= area of base and H= height
B= 11*7 = 77 feet²
H= 12 feet
Volume= 77*12=924 cubic feet
Cost =$3 per cubic foot
Total cost= 3*924= $2772
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Allan painted the circular patch on his driveway. He used the formula below to calculate the area of the circular patch. The diameter of the circular patch was 20 meters. What was the area of the patch? Assume pi=3.14
Answer: 314 square meters
Step-by-step explanation:
The formula for the area of a circle is given by A = πr^2, where r is the radius of the circle. Since the diameter of the circular patch is given as 20 meters, the radius would be half of that or 10 meters.
So, using the formula, we can calculate the area of the circular patch as follows:
A = πr^2
A = π(10)^2
A = 3.14(100)
A = 314 square meters
Therefore, the area of the circular patch is 314 square meters.
Please help!!!!!!!!!!!!!!
Step-by-step explanation:
try this option (see the attachment), answers are marked with red colour.
find a parameterization of each of the following surfaces, in terms of sines, cosines, and hyperbolic sines and cosines
Parameterizing a surface over a rectangle Parameterizing the surface z = x²+2y² over the rectangular region R defined by -3 ≤ x ≤ 3, −1 ≤ y ≤ 1 are falls under the range of R.
Let's start by expressing x and y as functions of u and v. Since x varies between -3 and 3 over R, we can use the following parameterization for x:
x = u
where u varies between -3 and 3. Similarly, since y varies between -1 and 1 over R, we can use the following parameterization for y:
y = v
where v varies between -1 and 1.
Next, we can use these parameterizations for x and y to express z as a function of u and v. Substituting x = u and y = v into the equation z = x² + 2y², we get:
z = u² + 2v²
So, the parameterization of the surface z = x² + 2y² over the rectangular region R is given by:
x = u, y = v, z = u² + 2v²
where -3 ≤ u ≤ 3 and -1 ≤ v ≤ 1.
The parameterization allows us to study various properties of the surface z = x² + 2y² over the rectangular region R.
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Complete Question:
Parameterizing a surface over a rectangle Parameterizing the surface z = x²+2y² over the rectangular region R defined by -3 ≤ x ≤ 3, −1 ≤ y ≤ 1.
Calculate the 90% confidence interval for the proportion of voters who cast their ballot for the candidate.
We can say with 90% confidence that the true proportion of voters who cast their ballot for the candidate lies between 0.564 and 0.636. We can calculate it in the following manner.
To calculate the 90% confidence interval for the proportion of voters who cast their ballot for the candidate, we need to use the following formula:
CI = p ± z√(p(1-p)/n)
where:
CI is the confidence interval
p is the sample proportion
z is the z-score corresponding to the desired confidence level (90% in this case)
n is the sample size
Assuming we have a sample of size n and a sample proportion of p who voted for the candidate, we need to find the value of z for the 90% confidence level. The z-score can be found using a z-table or a calculator, and for a 90% confidence level, the z-score is 1.645.
Substituting the values into the formula, we get:
CI = p ± 1.645√(p(1-p)/n)
For example, if the sample size is 1000 and the sample proportion is 0.6 (60% of voters voted for the candidate), then the 90% confidence interval would be:
CI = 0.6 ± 1.645√(0.6(1-0.6)/1000) = (0.564, 0.636)
Therefore, we can say with 90% confidence that the true proportion of voters who cast their ballot for the candidate lies between 0.564 and 0.636.
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Full question here:
Calculate the 90% confidence interval for the proportion of voters who cast their ballot for the candidate. Number of votes: 125
Voter Response Dummy Variable
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What is the average rate of change between
the points (17, 5) and (19, --1)?
The average rate of change between the points (17, 5) and (19, -1) is -3.
What is the average rate of change?If we have a given function y = f(x) with two known points (a, f(a)) and (b, f(b)), then the average rate of change in that interval [a, b] is:
R = ( f(b) - f(a))/(b - a)
Here we have the two points (17, 5) and (19, -1)
So we have:
a = 17 and f(a) = 5
b = 19 and f(b) = -1
Replacing that in the formula for the average rate of change we will get:
R = (-1 - 5)/(19 - 17)
R = -6/2
R = -3
The average rate of change is -3
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△CDE∼△PQR. CD=9 m, EC=15 m, PQ=15 m. What is the length of RP?
Answer:
RP = 25
Step-by-step explanation:
since the triangles are similar then the ratios of corresponding sides are in proportion, that is
[tex]\frac{RP}{EC}[/tex] = [tex]\frac{PQ}{CD}[/tex] ( substitute values )
[tex]\frac{RP}{15}[/tex] = [tex]\frac{15}{9}[/tex] ( cross- multiply )
9 RP = 15 × 15 = 225 ( divide both sides by 9 )
RP = 25
If a drug has a concentration of 5.315 mg per 3.743 mL, how many mL are needed to give 4.719 gram of the drug? Round to 1 decimal.
Answer:
888.4 mL.
Step-by-step explanation:
To solve this problem, we can use the following formula:
Amount of drug (in mg) = concentration (in mg/mL) × volume (in mL)
We are given the concentration of the drug as 5.315 mg per 3.743 mL. To find the volume of the drug needed to give 4.719 g, we need to rearrange the formula to solve for volume:
Volume (in mL) = amount of drug (in mg) ÷ concentration (in mg/mL)
First, we need to convert 4.719 g to mg by multiplying by 1000:
4.719 g × 1000 mg/g = 4719 mg
Now we can substitute the given concentration and the calculated amount of drug into the formula and solve for volume:
Volume (in mL) = 4719 mg ÷ 5.315 mg/mL
Volume (in mL) ≈ 888.5 mL
Therefore, approximately 888.5 mL of the drug are needed to give 4.719 g. Rounded to 1 decimal, the answer is 888.4 mL.
Please help me answer this question ASAP!!
Will mark as brainliest if correct and 50+ points!
Answer:
See explanation below
Step-by-step explanation:
1. 12x - 18 = 6(2x -3)
2. 15x + 25 = 5(3x + 5)
3. 14x + 21 = 7(2x + 3)
4. 5x - 5 = 5(x - 1)
5. 12x - 30 = 6(2x - 5)
6. 10x + 8 = 2(5x + 4)
7. 27x + 18 = 9(3x + 2)
8. 4x - 20 = 4(x - 5)
9. 20x + 30 = 10(2x + 3)
10. 4(x + 5) = 4x + 20
11. 3(x - 2) = 3x - 6
12. 5(2x + 4) = 10x + 20
13. 5(x - 1) = 5x - 5
14. 1/2(10x + 12) = 5x + 6
15. 4(2x + 4) = 8x + 16
16. 2(5x - 2) = 10x - 4
17. 2(x - 8) = 2x - 16
18. 4(2x + 1) = 8x + 4
Work out the value of the missing angle
x
.
The diagram is not drawn to scale.
Answer:
No diagram provided here
Bradley went to the store to buy ingredients for a new recipe. Artichokes were on sale for $3 per pound.
How much did Bradley pay if he bought
2
3
of a pound?
A $6. B $5. C $3 D $2
Answer :
Step-by-step explanation to problem:
2/3 * 3 = 2
we do 2/3 times 3 because $3 is for 1 pound and here we only need 2/3 of a pound
$2
Correct Answer = D
A water cooler springs a leak and empties in 2 minutes. The graph below shows the rate at which water leaks from the cooler as a function of time.
The amount of water that was in the cooler before it started leaking was 6 gallons.
Describe Integration?Integration is a mathematical process that involves finding the integral of a function. It is the reverse operation of differentiation, which involves finding the derivative of a function. The integral of a function is a measure of the area under the curve of the function, between two given limits of integration.
The graph shows the rate at which water leaks from the cooler as a function of time, which means that the y-axis represents the rate of leakage in gallons per minute (gal/min), and the x-axis represents the time in minutes.
Since we know that the cooler emptied in 2 minutes, we can integrate the leakage rate over the time interval [0, 2] to find the total amount of water that leaked out:
Total amount of water leaked = ∫[0,2] leakage rate(t) dt
The leakage rate is given by the graph, which consists of a straight line connecting two points: (0,6) and (2,0). We can express this line as a linear equation in slope-intercept form:
leakage rate(t) = mt + b
where m is the slope of the line and b is the y-intercept. To find the slope, we can use the formula:
m = (y2 - y1) / (x2 - x1)
where (x1, y1) = (0,6) and (x2, y2) = (2,0). Plugging in the values, we get:
m = (0 - 6) / (2 - 0) = -3
So the equation of the line is:
leakage rate(t) = -3t + 6
Now we can integrate this equation over the time interval [0, 2] to get the total amount of water leaked:
Total amount of water leaked = ∫[0,2] (-3t + 6) dt
= [-3t²/2 + 6t] from 0 to 2
= (-3(2)²/2 + 6(2)) - (-3(0)²/2 + 6(0))
= (6 - 0) - (0 - 0)
= 6 gallons
Therefore, the amount of water that was in the cooler before it started leaking was 6 gallons.
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The complete question is :
HEEELLLLPPPPP MEEEEEEEEE
1. Solve.
a. 2/5t = 6
b. -4.5 = a-8
c. 1/2+p=-3
d. 1/2 = x3
e. -12 = -3y
The equation is saying that -12 is equal to -3 multiplied by y. To solve for y, divide both sides by -3. This would give an answer of 4.
What is equation?An equation is a mathematical statement that expresses the equality or inequality of two values or expressions. It consists of two expressions connected by an equals sign, inequality sign or other relational operator. Equations can involve numbers, variables, and operations such as addition, subtraction, multiplication, division and exponentiation. An equation can be used to solve problems related to mathematics, science, engineering, finance, and many other disciplines. Equations can also be used to model and describe real-world phenomena.
t = 30
a = 12.5
p = -5.5
x = 2/3
y = 4.
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a. t = 30/2; To solve this equation, divide both sides by 2/5. The resulting equation is t = 30/2.
What is equation?An equation is a mathematical statement that expresses the equality of two expressions by using an equals sign (=). It states that the two expressions on either side of the equals sign are equal in value. An equation is an example of a mathematical problem, which can be used to solve real-world problems.
b. a = 4.5; To solve this equation, add 8 to both sides. The resulting equation is a = 4.5.
c. p = -7/2; To solve this equation, add 3 to both sides. The resulting equation is p = -7/2.
d. x = 2; To solve this equation, divide both sides by 3. The resulting equation is x = 2.
e. y = 4; To solve this equation, divide both sides by -3. The resulting equation is y = 4.
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PLS HELP FAST + BRAINLIEST!!
let z=a+bi/a-bi where a and b are real numbers. prove that z^2+1/2z is a real number.
Answer:
Step-by-step explanation:
To prove that z^2 + 1/2z is a real number, we need to show that the imaginary part of z^2 + 1/2z is equal to zero.
We know that z = (a+bi)/(a-bi)
Multiplying the numerator and denominator by the complex conjugate of the denominator, we get
z = (a+bi)(a+bi)/(a-bi)(a+bi)
z = (a^2 + 2abi - b^2)/(a^2 + b^2)
Expanding z^2, we get:
z^2 = [(a^2 + 2abi - b^2)/(a^2 + b^2)]^2
z^2 = (a^4 + 2a^2b^2 + b^4 - 2a^2b^2 + 4a^2bi - 4b^2i)/(a^4 + 2a^2b^2 + b^4)
Simplifying, we get:
z^2 = (a^4 - b^4 + 2a^2bi)/(a^4 + 2a^2b^2 + b^4)
Now, let's compute z^2 + 1/2z:
z^2 + 1/2z = (a^4 - b^4 + 2a^2bi)/(a^4 + 2a^2b^2 + b^4) + 1/2[(a+bi)/(a-bi)]
To simplify this expression, we need to find a common denominator:
z^2 + 1/2z = (2a^5 - 2a^3b^2 + 3a^4b - 3ab^4 - 2b^5 + 3a^3bi + 3ab^3i)/(2(a^4 + 2a^2b^2 + b^4))
We can see that the imaginary part of z^2 + 1/2z is (3a^3b - 3ab^3)/(2(a^4 + 2a^2b^2 + b^4))
However, we know that a and b are real numbers, so the imaginary part of z^2 + 1/2z is zero.
Therefore, z^2 + 1/2z is a real number.
the car drives at an average speed of 106 km per hour for 2 hours for 45 minutes at which constant speed must the car drive to travel the same distance in 2 hours 35 minutes
The car must drive at a constant speed of approximately 112.89 km/hr to cover the same distance in 2 hours 35 minutes.
What is the formula for Time?The formula for time is: time = distance / speed
where "distance" is the distance traveled by an object, and "speed" is the rate at which the object is moving.This formula can be used to calculate the time taken by an object to travel a certain distance at a constant speed, or to calculate the speed or distance if the other two variables are known.
What is the formula for Speed?The formula for speed is: speed = distance / time
where "distance" is the distance traveled by an object and "time" is the duration of travel.
This formula can be used to calculate the speed of an object if the distance it has traveled and the time it took to travel that distance are known. It can also be used to calculate the distance traveled by an object if its speed and the time it traveled at that speed are known.
In the given question,
Let's first calculate the distance traveled in 2 hours 45 minutes (2.75 hours) at an average speed of 106 km/hr.
distance = speed × time
distance = 106 × 2.75
distance = 291.5 km
Now, we need to find at which constant speed the car must drive to cover the same distance in 2 hours 35 minutes (2.5833 hours). Let's call this speed "x".
distance = speed × time
291.5 = x × 2.5833
x = 291.5 / 2.5833
x ≈ 112.89 km/hr
Therefore, the car must drive at a constant speed of approximately 112.89 km/hr to cover the same distance in 2 hours 35 minutes.
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What is the meaning of "invertible n x n matrices"?
Answer: A matrix A of dimension n x n is called invertible if and only if there exists another matrix B of the same dimension, such that AB = BA = I, where I is the identity matrix of the same order.
Step-by-step explanation:
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Use implicit differentiation to find an equation of the tangent line to the curve sin(x+y)=8x−8y at the point (π,π)
The equation of the tangent line to the curve sin( x y) = 8x- 8y on the factor( π, π) is y = (7/9) x-( 2π/ 9).
To discover the equation of the tangent line to the curve sin( x y) = 8x- 8y on the point( π, π), we want to apply implicit differentiation to discover the pitch of the tangent line at that point.
We begin through differencing both sides of the equation with reference to xcos( x y)( 1 dy/ dx) = eight- 8dy/ dx
After, we can simplify the expression by isolating the terms beholding dy/ dx on one aspect
cos( x y) cos( x y) dy/ dx = 8- 8dy/ dx
8 cos( x y)) dy/ dx = 8- cos( x y)
dy/ dx = ( 8- cos( x y))( 8 cos( x y))
Now we're suitable to discover the pitch of the tangent line at the factor( π, π) by plugging in x = π and y = π into the expression we simply derived
dy/ dx = ( 8- cos( 2π))( 8 cos( 2π))
dy/ dx = ( 8- 1)/( 8 1)
dy/ dx = 7/ nine
Thus, the pitch of the tangent line to the curve sin( x y) = 8x- 8y at the factor( π, π) is7/9.
To find the equation of the tangent line, we can use the point- slope form of the equation
y- y1 = m( x- x1)
In which m is the pitch we simply set up, and( x1, y1) is the point( π, π). Plugging in the values, we get
y- π = ( 7/ nine)( x- π)
Simplifying, we get
y = ( 7/ nine) x-( 2π/ nine)
Thus, the equation of the tangent line to the curve sin( x y) = 8x- 8y on the factor( π, π) is y = (7/9) x-( 2π/ 9).
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3 Open Ended Two fractions have a common denominator
of 8. What could the two fractions be?
3. what cou
two fractions with a common denominator of 8 can be expressed in the form of a/b and c/8, where a and c are integers. As long as a and c are not both multiples of 8 then these fractions would have a common denominator of 8.
What is common denominator ?A number that can be divided exactly by all of the denominators in a group of fractions is referred to as a common denominator. 2. A noun that counts. A trait or attitude that all members of a group share is known as a common denominator.
According to the given information:Since the two fractions have a common denominator of 8, they can be written in the form of a/b and c/8, where a and c are integers.
There are many possible combinations of integers that could satisfy this condition. Here are some examples:
1/8 and 3/8
2/8 (which simplifies to 1/4) and 6/8 (which simplifies to 3/4)
4/8 (which simplifies to 1/2) and 7/8
5/8 and 2/8 (which simplifies to 1/4)
3/8 and 4/8 (which simplifies to 1/2)
In general, any two fractions with a common denominator of 8 can be expressed in the form of a/b and c/8, where a and c are integers. As long as a and c are not both multiples of 8 then these fractions would have a common denominator of 8.
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In △ △ ABC, CJ = 18. If CG = BG, what is KJ? Triangle A B C is divided by 4 segments. A H is the height. C J extends from C to side A B. B I extends from B to side A C. H I extends from the height on B C to I on A C. C J and B I intersect at point K. A J and B J are congruent. A I and C I are congruent.
Solving for CI in terms of the given lengths, we get: [tex]Cl=\frac{\sqrt{BG^{2} -IM^{2} } }{\sqrt{2} }[/tex]
Substituting this expression for CI and the given value for CG into the expression for BI, we get: [tex]BI=CG-\frac{\sqrt{BG^{2} -IM^{2} } }{\sqrt{2} }[/tex].
What is triangle?A triangle is a three-sided polygon, which is a closed two-dimensional shape with straight sides. In a triangle, the three sides connect three vertices, or corners, and the angles formed by these sides are called the interior angles of the triangle. The sum of the interior angles of a triangle is always 180 degrees. Triangles can be classified by their side lengths and angle measurements. For example, an equilateral triangle has three sides of equal length, and all of its angles are 60 degrees; an isosceles triangle has two sides of equal length, and its base angles are also equal; a scalene triangle has three sides of different lengths, and all of its angles are also different. Triangles are a fundamental shape in mathematics and geometry, and they have numerous applications in fields such as architecture, engineering, physics, and more.
Given by the question.
Based on the given information, we can start by drawing a diagram of triangle ABC and the segments AH, BJ, CI, CJ, and BI as described.
Since CG = BG, we can draw the perpendicular bisector of side AC passing through point G, which will intersect side AB at its midpoint M.
Now, we can see that triangle CGB is isosceles with CG = BG, so the perpendicular bisector of side CB also passes through point G. This means that G is the circumcenter of triangle ABC, and therefore, the distance from G to any vertex of the triangle is equal to the radius of the circumcircle.
Next, we can use the fact that AJ and BJ are congruent to draw the altitude from point J to side AB, which we will call JN. Similarly, we can draw the altitude from point I to side BC, which we will call IM.
Since AJ and BJ are congruent, the altitude JN will also be the perpendicular bisector of side AB, so it will pass through point M. Similarly, the altitude IM will pass through point G, which is the circumcenter of triangle ABC.
Now, we can use the Pythagorean theorem to find the lengths of JN and IM in terms of the given lengths:
[tex]JN^{2}= AJ^{2} -AN^{2} \\ = ( AH+HN)^{2} - AN^{2} \\=AH^{2} +2AH*HN+HN^{2}-AN^{2} \\[/tex]
[tex]IM^{2}= CI^{2} -CM^{2} \\=( CG-GM)^{2} -CM^{2} \\CG^{2}-2CG*GM+GM^{2} -CM^{2}[/tex]
Since CG = BG and GM = BM (since M is the midpoint of AB), we can simplify the expression for IM^2 as follows:
[tex]IM^{2}[/tex] = [tex]BG^{2}[/tex] - 2BG * BM + [tex]BM^{2}[/tex] - [tex]CM^{2}[/tex]
= [tex]BG^{2}[/tex] - [tex]BM^{2}[/tex] - [tex]CM^{2}[/tex]
Now, we can use the fact that BJ and CI intersect at point K to find the length of KJ:
KJ = BJ - BJ * (CK/CI)
= BJ * (1 - CK/CI)
= BJ * (1 - BM/CM)
To find BM/CM, we can use the fact that triangle BCI is isosceles with BI = CI, so the altitude IM is also a median of the triangle. This means that CM = 2/3 * BI. Similarly, we can find BJ in terms of JN using the fact that triangle ABJ is isosceles with AJ = BJ:
BJ = 2 * JN
Substituting these expressions into the equation for KJ, we get:
KJ = 2 * JN * (1 - 2/3 * BI/CM)
Now, we just need to find BI/CM in terms of the given lengths. Using the fact that triangle BCI is isosceles with BI = CI, we can find BI in terms of CG:
BI = CG - CI
Substituting this expression into the equation for [tex]IM^{2}[/tex]and simplifying, we get:
[tex]IM^{2}[/tex] =[tex]BG^{2}[/tex] - CG * CI
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the values or variables listed in the function declaration are called _____ paramters to the function.
The values or variables listed in the function declaration are called formal parameters to the function.
They are used to store the data that is passed into the function when it is called. The formal parameters are local variables, meaning that the values stored in them are only available within the function.
The arguments are the values passed to the function when it is called. These values are then assigned to the formal parameters and are used within the function to perform the desired task.
Formal arguments are produced at function entry and removed at function exit, behaving similarly to other local variables inside the function.
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Can I please get help it's an EMERGENCY!
The number of hours it will take the same dog to run 26 1/10 miles is 7.2 hours
How long will it take the dog to run 26 1/10 miles?7 1/4 miles in 2 hours
26 1/10 miles in x hours
Equate miles ratio hours
7 ¼ miles : 2 hours = 26 ⅒ miles : x hours
7.25 / 2 = 26.10 / x
cross product
7.25 × x = 26.10 × 2
7.25x = 52.20
divide both sides by 7.25
x = 52.20 / 7.25
x = 7.2 hours
Ultimately, it will take 7.2 hours for the dog to run 26⅒ miles.
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Roberto must make his costume for the school play. He needs a piece of fabric that is 2 2/3 yards long and 1 1/2 yard wide. What is the area of the piece of fabric Roberto needs?
Roberto needs 4 square yards of fabric to make his costume.
What is improper fraction?A fraction that has the numerator higher than or equal to the denominator is said to be inappropriate. For instance, the fraction 7/3 is incorrect since 7 is bigger than 3. Mixed numbers, which combine a whole number and a correct fraction, can be created from improper fractions.
Given that, piece of fabric that is 2 2/3 yards long and 1 1/2 yard wide.
Convert the length from a mixed number to an improper fraction:
2 2/3 = (2 x 3 + 2)/3 = 8/3
1 1/2 = 3/2
The area of the rectangle is:
Area = Length x Width
Substituting the values we have:
Area = (8/3) x (3/2) = 4
Hence, Roberto needs 4 square yards of fabric to make his costume.
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Mark is going to an awards dinner and wants to dress appropriately. He is running behind schedule and asks his little brother to randomly select an outfit for him.
Mark has one blue dress shirt, one white dress shirt, one black dress shirt, one pair of black slacks, one pair of grey slacks, and one red tie. All six of his possible outfits are listed below.
Let
A
AA be the event that Mark's little brother selects an outfit with a white shirt and grey slacks and
B
BB be the event that he selects an outfit with a black shirt.
What is
P
(
A
or
B
)
P(A or B)P, left parenthesis, A, start text, space, o, r, space, end text, B, right parenthesis, the probability that Mark's little brother selects an outfit with a white shirt and grey slacks or an outfit with a black shirt?
6. Deepa's age is three times that of her brother Devan. After 2 years Deepa's age would
be two times that of Devan. How old are they now?
Answer:
Devan's age = 2 years.
Deepa's age = 6 years.
Step-by-step explanation:
Framing and solving algebraic equation:Present age:
Let the present age of Devan = x
Present age of Deepa = 3x
After 2 years:
Age of Devan = x + 2
Age of Deepa = 3x + 2
Deepa's age = 2* Devan's age
3x + 2 = 2 *(x + 2)
3x + 2 = 2x + 2*2 {Use distributive property}
3x + 2 = 2x + 4
Subtract '2' from both sides,
3x = 2x + 4 - 2
3x = 2x + 2
Subtract '2x' from both sides,
3x - 2x = 2
x = 2
Devan's age = 2 years.
Deepa's age = 3*2
= 6 years
Calculate the area of the shaded segments in the following diagrams. (a) 12 cm 40° (b) 58° 16 cm
(a) 12 cm 40° : Area of shaded segments = 301.44 sq. cm.
(b) 58° 16 cm : Area of shaded segments = 777.04 sq. cm.
Explain about the sector of circle?Two radii that meet at the center to form a sector define a circle. The sector is the portion of the circle created by these two radii. Knowing a circle's central angle calculation and radius measurement are both crucial for solving circle-related difficulties.
Area of sector of circle = Ф/360 * πr²
π = 3.14
r is the radius
Ф is the angle subtended.
(a) 12 cm 40°
Area of shaded segments = 40/60 * 3.14* 12²
Area of shaded segments = 40/60 * 452.16
Area of shaded segments = 301.44 sq. cm.
(b) 58° 16 cm
Area of shaded segments = 58/60 * 3.14* 16²
Area of shaded segments = 58/60 * 803.84
Area of shaded segments = 777.04 sq. cm.
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The diagram for the question is attached.
Let V and W be vector spaces and T: v → w be linear. (a) Prove that T is one-to-one if and only if T carries linearly inde- pendent subsets of V onto linearly independent subsets of W. (b) Suppose that T is one-to-one and that S is a subset of V. Prove that S is linearly independent if and only if T(S) is linearly inde- pendent. Suppose β and onto. Prove that T(3) = {T(m), T(v2), for W (c) (vi, v2 , . . . , Un} is a basis for V and T is one-to-one ,T(vn)} is a basis
(a) T is one-to-one if and only if T carries linearly independent subsets of V onto linearly independent subsets of W.
(b) If T is one-to-one, then S is linearly independent if and only if T(S) is linearly independent.
(c) If β is a basis for V and T is one-to-one and onto, then T(β) is a basis for W.
(a) Assume T is one-to-one. Let S be a linearly independent subset of V, and suppose T(S) is linearly dependent. Then there exist distinct vectors s1, s2, ..., sn in S such that T(s1), T(s2), ..., T(sn) are linearly dependent. This means that there exist scalars c1, c2, ..., cn, not all zero, such that c1T(s1) + c2T(s2) + ... + cnT(sn) = 0. Since T is linear, we have T(c1s1 + c2s2 + ... + cnsn) = 0. But since T is one-to-one, this implies that c1s1 + c2s2 + ... + cnsn = 0, contradicting the assumption that S is linearly independent. Hence, T(S) must be linearly independent.
Conversely, assume that T carries linearly independent subsets of V onto linearly independent subsets of W. Let v1 and v2 be distinct vectors in V, and suppose T(v1) = T(v2). Then {v1, v2} is linearly dependent, which implies that there exist scalars c1 and c2, not both zero, such that c1v1 + c2v2 = 0. Applying T to both sides yields c1T(v1) + c2T(v2) = 0, which implies that T(v1) and T(v2) are linearly dependent. This contradicts the assumption that T carries linearly independent subsets of V onto linearly independent subsets of W. Hence, T must be one-to-one.
(b) Assume T is one-to-one and let S be a subset of V. Suppose S is linearly independent and that T(S) is linearly dependent. Then there exist distinct vectors s1, s2, ..., sn in S such that T(s1), T(s2), ..., T(sn) are linearly dependent. This means that there exist scalars c1, c2, ..., cn, not all zero, such that c1T(s1) + c2T(s2) + ... + cnT(sn) = 0. Since T is linear, we have T(c1s1 + c2s2 + ... + cnsn) = 0. But since T is one-to-one, this implies that c1s1 + c2s2 + ... + cnsn = 0, contradicting the assumption that S is linearly independent. Hence, T(S) must be linearly independent.
Conversely, assume that T(S) is linearly independent whenever S is a linearly independent subset of V. Let v1 and v2 be distinct vectors in V, and suppose T(v1) = T(v2). Then {v1, v2} is linearly dependent, which implies that there exist scalars c1 and c2, not both zero, such that c1v1 + c2v2 = 0. Since {v1, v2} is linearly dependent, we have either v1 = 0 or v2 = 0. Without loss of generality, assume v1 = 0. Then T(v1) = 0 = T(v2), and hence T({v1, v2}) = {0} is linearly dependent. This contradicts the assumption that T carries linearly independent subsets of V onto linearly independent subsets of W. Hence, S must be linearly independent.
(c) First, we will show that T(β) spans W. Let w be an arbitrary vector in W. Since T is onto, there exists some vector v in V such that T(v) = w. Since β is a basis for V, there exist scalars c1, c2, ..., cn such that v = c1v1 + c2v2 + ... + cnvn. Applying T to both sides, we have w = T(v) = T(c1v1 + c2v2 + ... + cnvn) = c1T(v1) + c2T(v2) + ... + cnT(vn), which implies that T(β) spans W.
Next, we will show that T(β) is linearly independent. Suppose there exist scalars c1, c2, ..., cn such that c1T(v1) + c2T(v2) + ... + cnT(vn) = 0. Applying T to both sides, we have T(c1v1 + c2v2 + ... + cnvn) = 0. But since T is one-to-one, this implies that c1v1 + c2v2 + ... + cnvn = 0, which implies that c1 = c2 = ... = cn = 0, since β is a basis for V. Hence, T(β) is linearly independent.
Since T(β) spans W and is linearly independent, it is a basis for W. Therefore, if β is a basis for V and T is one-to-one and onto, then T(β) is a basis for W.
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