(AMC8, 1994) Each of the three large squares shown below is the same size. Segments that intersect the sides of the squares intersect at midpoints of the sides. How do the areas of the shaded regions compare?

A. The shaded areas in all three are equal.
B. Only the shaded areas I and II are equal.
C. Only the shaded areas of I and III are equal.
D. Only the shaded areas of II and III are equal.
E. The shaded areas of I, II and III are all different.

3⁴ is 3 times larger than 3³, 5³ is 5 times larger than 5², 7¹⁰ is 49 times larger than 7⁸ and 17⁶ is 289 times larger than 17⁴.

3⁴ / 3³ = (3 × 3 × 3 × 3) / (3 × 3 × 3) = 3

This means that 3⁴ is 3 times larger than 3³.

5³ is 5 times larger than 5².

5³ / 5² = (5 × 5 × 5) / (5× 5) = 5

7¹⁰ is 49 times larger than 7⁸.

7¹⁰/ 7⁸ = 7² =49

17⁶ is 289 times larger than 17⁴.

17⁶ /17⁴ = 289

Hence, 3⁴ is 3 times larger than 3³, 5³ is 5 times larger than 5², 7¹⁰ is 49 times larger than 7⁸ and 17⁶ is 289 times larger than 17⁴.

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Answer:

tan Ф = opposite / adjacent = a / b=5/12

Step-by-step explanation:

sec Ф = hypotenuse / adjacent = c / b=13/12

tan Ф = opposite / adjacent = a / b

find a

a^2+b^2=c^2

a=√c^2-b^2 b=12 and c=13

a=√144-169=√25=5

\(\text{To get the total surface area, all we have to do is multiply } 18 \text{ by } 12, \text{which gets us}\)\($18\cdot12 = \boxed{216\text{ ft}^2}\).

\(\text{So, our answer is } \boxed{216\text{ ft}^2}.\)

Larry needs 216 square feet of carpeting for her rectangular living room.

To find the amount of carpeting Larry needs, we need to calculate the area of her rectangular living room. The area of a rectangle can be found by multiplying its length by its width. In this case, the length of the living room is 18 feet and the width is 12 feet.

So, the area of the living room is:

Area = Length * Width

Area = 18 feet * 12 feet

Area = 216 square feet

Therefore, Larry needs 216 square feet of carpeting for her living room.

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Answer:

3x + 8y = 0

Step-by-step explanation:

µ > 0: θ(x,t) = A sin(ωt – kx) + B cos(ωt – kx). µ = 0: θ(x,t) = At + B. µ < 0: ∂²θ/∂t² + |µ|/L² ∂²θ/∂x² = 0

Case 1: µ > 0
In this case, when µ > 0, the equation governing the displacement θ(x,t) is given by the wave equation:
∂²θ/∂t² - µ/L² ∂²θ/∂x² = 0
The general solution to this wave equation is:
Θ(x,t) = A sin(ωt – kx) + B cos(ωt – kx)
Where A and B are constants, ω is the angular frequency, and k is the wave number. The angular frequency ω and the wave number k are related as ω = v * k, where v is the wave velocity. In this case, the wave velocity is given by v = sqrt(µ/L²).
Case 2: µ = 0
When µ = 0, the equation governing the displacement θ(x,t) simplifies to:
∂²θ/∂t² = 0
This equation indicates that there is no wave-like behavior in the system. The general solution in this case is:
Θ(x,t) = At + B
Where A and B are constants determined by the initial conditions.
Case 3: µ < 0
When µ < 0, the equation governing the displacement θ(x,t) becomes:
∂²θ/∂t² + |µ|/L² ∂²θ/∂x² = 0
The general solution to this equation can be expressed as a combination of sine and hyperbolic sine functions. However, without specific initial conditions, it is not possible to provide a detailed solution.

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Answer:

70

Step-by-step explanation:

Evaluate 3 b^2 - b where b = 5:

3 b^2 - b = 3×5^2 - 5

Hint: | Evaluate 5^2.

5^2 = 25:

3×25 - 5

Hint: | Multiply 3 and 25 together.

3×25 = 75:

75 - 5

Hint: | Subtract 5 from 75.

| 7 | 5

- | | 5

| 7 | 0:

Answer:  70

Answer: i hope this will help you

a^x=y this is a simple equation

where as loga(in the base) y=x this is a logarithmic equation

1. Identify the curve and the point: The curve is y = sec(x), and the point is (π/6, 2√3/3).

2. Calculate the derivative of the curve: The derivative of y = sec(x) is y' = sec(x)tan(x).

3. Evaluate the derivative at the given point: At x = π/6, we have y' = sec(π/6)tan(π/6). Since sec(π/6) = 2 and tan(π/6) = √3/3, we get y' = 2(√3/3) = 2√3/3.(4). Use the point-slope form of the line equation: y - y1 = m(x - x1), where (x1, y1) is the given point and m is the slope (derivative).

5. Plug in the values: y - (2√3/3) = (2√3/3)(x - π/6). This is the equation of the tangent line to the curve y = sec(x) at the point (π/6, 2√3/3).

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Thus we only take the positive root:x = 27/8 = 3.375Answer: 3.375 feet.

The problem states that a rectangular garden that measures 50 feet wide and 45 feet long is enclosed by a uniform width of x feet paved path. To solve the problem,

we can use the formula of the combined area of the garden and the paved path and equate it to 2646 square feet. The combined area is computed by adding the area of the garden and the area of the paved path.

Garden area:Length of garden = 45 ftWidth of garden = 50 ftArea of garden = Length x Width= 45 x 50= 2250 square feet

Paved path:

If the garden has a uniform width of x feet paved path, then the width of the paved path would be x + 2x + x= 4x. The width is multiplied by 2 because there are two widths surrounding the garden.

Length of paved path = length of garden + 2 (width of paved path)= 45 + 2 (4x)= 8x + 45Width of paved path = width of garden + 2 (width of paved path)= 50 + 2 (4x)= 8x + 50

The area of the paved path is computed by subtracting the area of the garden from the combined area.Area of paved path = Combined area - Garden area2646 square feet

= (8x + 45) (8x + 50) - 2250= 64x² + 760x + 675

We then solve for the value of x by factoring the quadratic equation.2646 square feet = 64x² + 760x + 6752646 - 2646

= 64x² + 760x + 675 - 264664x² + 760x - 1971

= 0(8x - 27) (8x + 73) = 0

The value of x cannot be negative,

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Answer:

v= 3

Step-by-step explanation:

hope this helps!

Answer:

V = 3

Step-by-step explanation:

4(v + 2) = 20               we must first distribute

4v + 8 = 20                 we subtract 8 from both sides

4v + 8 - 8 = 20 - 8      we simplify that

4v = 12                        we divide both sides by 12 to isolate v

V = 3

Answer:

The given line segment has a midpoint at (−1, −2).

On a coordinate plane, a line goes through (negative 5, negative 3), (negative 1, negative 2), and (3, negative 1).

What is the equation, in slope-intercept form, of the perpendicular bisector of the given line segment?

y = −4x − 4

y = −4x − 6

y = One-fourthx – 4

y = One-fourthx – 6

y = Three-halvesx + 1

Both -5 and 1 are extraneous solution to the equation

A quadratic equation is a second-order polynomial equation in a single variable x ax2+bx+c=0. with a ≠ 0 . Because it is a second-order polynomial equation, the fundamental theorem of algebra guarantees that it has at least one solution. The solution may be real or complex.

c²+2c-4-1-2c = 0

c²+2c-2c -4-1 = 0

c²-5 = 0

Therefore both -5 and 1 are extraneous solution in the equation.

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Answer:

#3 is 100

#2 is west

#1 is 80

Step-by-step explanation:

The values of x in the interval [0, 2π] that satisfy the equation 8sin(2x) = 4 are π/12 and 5π/12.

To find the values of x that satisfy the equation 8sin(2x) = 4 in the interval [0, 2π], we can solve for x by isolating sin(2x) first and then finding the corresponding angles.

Let's solve the equation step by step:

8sin(2x) = 4

Divide both sides of the equation by 8:

sin(2x) = 4/8

sin(2x) = 1/2

To find the values of x, we need to determine the angles whose sine is 1/2. These angles occur in the first and second quadrants.

In the first quadrant, the reference angle whose sine is 1/2 is π/6.

In the second quadrant, the reference angle whose sine is 1/2 is also π/6.

However, since we're dealing with 2x, we need to consider the corresponding angles for π/6 in each quadrant.

In the first quadrant, the corresponding angle is π/6.

In the second quadrant, the corresponding angle is π - π/6 = 5π/6.

Now, let's find the values of x in the interval [0, 2π] that satisfy the equation:

For the first quadrant:

2x = π/6

x = π/12

For the second quadrant:

2x = 5π/6

x = 5π/12

Therefore, the values of x in the interval [0, 2π] that satisfy the equation 8sin(2x) = 4 are π/12 and 5π/12.

So, the comma-separated list of values is π/12, 5π/12.

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The net change in the value of the function between x = 1 and x = 6 is 30.

To find the net change in the value of the function between the inputs of 1 and 6, we need to find the difference between the output values of the function at x = 1 and x = 6, and then take the absolute value of that difference.

First, we can find the output value of the function at x = 1:

f(1) = 6(1) - 5 = 1

Next, we can find the output value of the function at x = 6:

f(6) = 6(6) - 5 = 31

The net change in the value of the function between x = 1 and x = 6 is the absolute value of the difference between these two output values:

|f(6) - f(1)| = |31 - 1| = 30

Therefore, the net change in the value of the function between x = 1 and x = 6 is 30.

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Since none of the 13 classmates had been given math homework between the original survey and Kelly's second survey, the sum of the values in the second data set is the same as the sum of the values in the original data set. Therefore, the change in the means can be determined without calculating the mean of either data set by considering the number of data points in each set.

Since both data sets have the same number of data points, the change in the means will be zero. This is because the mean is calculated by dividing the sum of the values by the number of data points, and since the sum of the values is the same in both data sets, the means will also be the same.

In other words, the change in the mean is calculated as follows:

Change in mean = Mean of second data set - Mean of first data set

Since none of the values in the second data set have changed, the mean of the second data set is the same as the mean of the first data set. Therefore, the change in the mean is:

Change in mean = Mean of second data set - Mean of first data set

= Mean of first data set - Mean of first data set

= 0

Thus, the change in the means between Kelly's original survey and her second survey is zero.

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Answer:

A.

Step-by-step explanation:

sqrt(a/b) = sqrt(a)/sqrt(b)

sqrt(2/9) = sqrt(2)/sqrt(9) = sqrt(2)/3

9514 1404 393

Answer:

  D

Step-by-step explanation:

The coefficient of 5/4 in the given equation tells you that you are looking for a table or graph that has a y-increase of 5 for each x-increase of 4.

Graph D is the one.

__

Let's look at the choices:

  A: y increases for an x-increase of 4

  B. y increases 16 (from -7 to 9) for an x-increase of 4 (from -2 to 2)

  C. y increases -5 (from -9 to -14) for an x-increase of 4 (from 4 to 8)

  D. y increases 5 (from -2 to 3) for an x-increase of 4 (from -4 to 0)

Answer:

The answer is 12860082/100000000000

The fraction form of the expression is -1/7776.

An expression contains one or more terms with addition, subtraction, multiplication, and division.

We always combine the like terms in an expression when we simplify.

We also keep all the like terms on one side of the expression if we are dealing with two sides of an expression.

Example:

1 + 3x + 4y = 7 is an expression.

3 + 4 is an expression.

2 x 4 + 6 x 7 – 9 is an expression.

33 + 77 – 88 is an expression.

We have,

\((-1/6)^5\)

This can be simplified as,

= \(\frac{(-1)^5}{6^5}\)

= -1/7776

Thus,

The fraction form of the expression is -1/7776.

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Sarah can serve a maximum of 6 people at Thanksgiving dinner with the 2 3/5 pounds of potatoes she bought.

To find out how many people Sarah can serve at Thanksgiving dinner, we need to divide the total amount of potatoes she bought by the amount each person will eat.

Sarah bought 2 3/5 pounds of potatoes.

To convert this mixed number to an improper fraction, we multiply the whole number (2) by the denominator (5) and add the numerator (3), giving us 13/5 pounds.

Each person will eat 3/8 of a pound of potatoes.

To find out how many people can be served, we divide the total amount of potatoes by the amount each person will eat:

(13/5) ÷ (3/8) = (13/5) * (8/3) = 104/15

So, Sarah can serve approximately 104/15 people.

Since we are looking for a whole number of people, we need to round down to the nearest whole number.

Therefore, Sarah can serve a maximum of 6 people at Thanksgiving dinner.

Sarah can serve a maximum of 6 people at Thanksgiving dinner with the 2 3/5 pounds of potatoes she bought.

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Part 1: Kate should buy 100 coats to maximize profits.Part 2: The probability that the coats sell out is 0.25 (25%), and the probability that they do not sell out is 0.75 (75%).

To maximize profits, Kate should consider the scenario where she sells all the coats without any left over at the end of the season.

Since the sales are uniformly distributed between 100 and 400, buying 100 coats ensures that she meets the minimum expected sales of 100. Purchasing more than 100 coats would increase costs without a guarantee of higher sales, potentially leading to excess inventory and lower profits.

Given that the sales are uniformly distributed between 100 and 400 coats, Kate's purchase of 100 coats covers the minimum expected sales.

The probability of selling out can be calculated by finding the proportion of the range covered by the desired sales (100 out of 300). Therefore, the probability of selling out is 100/300 = 0.25 or 25%. The probability of not selling out is the complement, which is 1 - 0.25 = 0.75 or 75%.

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If the function f(x) = |x-5| + 2, then the value of f(3) is 4

The function

f(x) = |x-5| + 2

The function is defined as the mathematical statement that shows the relationship between the independent variable and the dependent variable. The function consist of different variables, numbers and mathematical operators

Here the function consist of the absolute value symbol.

|-a| = a

The absolute value of the positive and the negative number will be always positive.

The function is f(x) = |x-5| + 2

Then,

f(3) = |3-5| + 2

= |-2| + 2

= 2 + 2

= 4

Therefore, the value of f(3) is 4

I have answered the question in general, as the given question is incomplete

The complete question is:

If f(x)=|x−5| + 2, find the value of f(3).

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To evaluate the line integral, we need to parametrize the given curve C and then substitute the parametric equations into the integrand. We can parameterize C using two line segments as follows:

For the first line segment from (0, 0) to (4, 0), we can let x = t and y = 0, where 0 ≤ t ≤ 4.

For the second line segment from (4, 0) to (5, 2), we can let x = 4 + t/√5 and y = 2t/√5, where 0 ≤ t ≤ √5.

Then the line integral becomes:

∫C xy dx +(x - y)dy = ∫0^4 t(0) dt + ∫0^√5 [(4 + t/√5)(2t/√5) dt + (4 + t/√5 - 2t/√5)(2/√5) dt]

Simplifying the integrand, we get:

∫C xy dx +(x - y)dy = ∫0^4 0 dt + ∫0^√5 [(8/5)t^2/5 + (8/5)t - (2/5)t^2/5 + (8/5)] dt

Evaluating the definite integral, we get:

∫C xy dx +(x - y)dy = [(8/25)t^5/5 + (4/5)t^2/2 + (8/5)t]0^√5 + [(2/25)t^5/5 + (4/5)t^2/2 + (8/5)t]0^√5

Simplifying, we get:

∫C xy dx +(x - y)dy = (16/5)(√5 - 1)

Therefore, the value of the line integral is (16/5)(√5 - 1).

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The smallest value of n is  mathematically given as

x'=1.66

This is further explained below.

Generally, the worth may mean a number of different things, all of which are closely connected. Any finite mathematical object may be a mathematical value. This is often a number in basic mathematics, whether it a real number like or an integer like 42.

In conclusion, to determine this we take a common value of n for all expression

let n be 5

Therefore

x=36+n,

x=36+5

x=40

x'=n/3

x'=5/3

x'=1.66

x=√n+16

x=√5+16

x=18.236067

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Answer:

v=20π ft/s

Step-by-step explanation:

Given:

Distance from the security car to the movie theater, D=25 ft

Distance of the reflection from the car, d=30 ft

Speed of rotation of the strobe lights, 2 rev/s

To find the speed at which the reflection of the security strobe lights is moving along the wall of the movie theater, we need to calculate the linear velocity of the reflection when it is 30 ft from the car.

We can start by finding the angular velocity in radians per second. Since the strobe lights rotate at 2 revolutions per second, we can convert this to radians per second.

ω=2πf

=> ω=2π(2)

=> ω=4π rad/s

The distance between the security car and the reflection on the wall of the theater is...

r=30-25= 5 ft

The speed of reflection is given as (this is the linear velocity)...

v=ωr

Plug our know values into the equation.

v=ωr

=> v=(4π)(5)

∴ v=20π ft/s

Thus, the problem is solved.

The speed of the reflection of the security strobe lights along the wall of the movie theater is 2π ft/s.

To solve this problem, we can use the concept of related rates. Let's consider the following variables:

x: Distance between the security car and the movie theater wall

y: Distance between the reflection of the security strobe lights and the security car

θ: Angle between the line connecting the security car and the movie theater wall and the line connecting the security car and the reflection of the strobe lights

We are given:

x = 25 ft (constant)

y = 30 ft (changing)

θ = 2 revolutions per second (constant)

We need to find the speed at which the reflection of the security strobe lights is moving along the wall (dy/dt) when the reflection is 30 ft from the car.

Since we have a right triangle formed by the security car, the movie theater wall, and the reflection of the strobe lights, we can use the Pythagorean theorem:

x^2 + y^2 = z^2

Differentiating both sides of the equation with respect to time (t), we get:

2x(dx/dt) + 2y(dy/dt) = 2z(dz/dt)

Since x is constant, dx/dt = 0. Also, dz/dt is the rate at which the angle θ is changing, which is given as 2 revolutions per second.

Plugging in the known values, we have:

2(25)(0) + 2(30)(dy/dt) = 2(30)(2π)

Simplifying the equation, we find:

60(dy/dt) = 120π

Dividing both sides by 60, we get:

dy/dt = 2π ft/s

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Answer:

s = - 12.8

Step-by-step explanation:

Given

\(\frac{s}{4}\) = - 3.2 ( multiply both sides by 4 to clear the fraction )

s = 4 × - 3.2 = - 12.8

Answer:

s = -12.8

Step-by-step explanation:

1. First we need to simplify both sides of the equation.

1/4s = -3.2

2. Now we need to multiply both sides by 4.

4 × ( 1/4s ) = ( 4 ) × ( −3.2 )

s = −12.8

the probability that the diameter of a elected bearing is greater than 85 millimeter P(diameter > 85) = P(z > (85-81)/4) = P(z > 1) = 0.1587

The diameter of ball bearings is normally distributed, with a mean of 81 millimeters and a variance of 16.

To calculate the probability that a selected bearing has a diameter greater than 85 millimeters, we first calculate the z-score for 85 millimeters.

We subtract 81 from 85 to get 4, and divide by 4 to get 1 for the z-score.

We the look up the probability for a value of 1 in the z-table, which is 0.1587.

This is the probability that a selected bearing has a diameter greater than 85 millimeters, rounded to four decimal places.

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Determine the height of television by using the pythagoras theorem.

So height of the television is 10.75 inch.