Rectangles - High School Math

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Question

Two rectangles are similar. The perimeter of the first rectangle is 36. The perimeter of the second is 12. If the base length of the second rectangle is 4, what is the height of the first rectangle?

Answer

Solve for the height of the second rectangle.

Perimeter = 2B + 2H

12 = 2(4) + 2H

12 = 8 + 2H

4 = 2H

H = 2

If they are similar, then the base and height are proportionally equal.

B1/H1 = B2/H2

4/2 = B2/H2

2 = B2/H2

B2 = 2H2

Use perimeter equation then solve for H:

Perimeter = 2B + 2H

36 = 2 B2 + 2 H2

36 = 2 (2H2) + 2 H2

36 = 4H2 + 2 H2

36 = 6H2

H2 = 6

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Question

Mark is making a plan to build a rectangular garden. He has 160 feet of fence to form the outside border of the garden. He wants the dimensions to look like the plan outlined below:

What is the area of the garden, rounded to the nearest square foot?

Answer

Perimeter: Sum of the sides:

4x + 4x + 2x+8 +2x+8 = 160

12x + 6 = 160

12x = 154

x = $\frac{77}{6}$

Therefore, the short side of the rectangle is going to be:

$\dpi{100} \dpi{100} 2\cdot \frac{77}{6}+3=\frac{172}{6}=\frac{86}{3}$

And the long side is going to be:

$\dpi{100} 4\cdot \frac{77}{6}=\frac{308}{6}=\frac{154}{3}$

The area of the rectangle is going to be as follows:

Area = lw

$\dpi{100} Area = \frac{86}{3}\cdot \frac{154}{3}=\frac{13,244}{9}=1471.55\approx 1472\ ft^{2}$

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Question

Erin is getting ready to plant her tulip garden. She wants to plant two tulips per square foot of garden. If her rectangular garden is enclosed by 24 feet of fencing, and the length of the fence is twice as long as its width, how many tulips will Erin plant?

Answer

We know that the following represents the formula for the perimeter of a rectangle:

In this particular case, we are told that the length of the fence is twice as long as the width. We can write this as the following expression:

Use this information to substitute in a variable for the length that matches the variable for width in our perimeter equation.

We also know that the length is two times the width; therefore, we can write the following:

$l = 2\times4$

The area of a rectangle is found by using this formula:

$A=8\times4$

The area of the garden is 32 square feet. Erin will plant two tulips per square foot; thus, she will plant 64 tulips.

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Question

A contractor is going to re-tile a rectangular section of the kitchen floor. If the floor is 6ft x 3ft, and he is going to use square tiles with a side of 9in. How many tiles will be needed?

Answer

We have to be careful of our units. The floor is given in feet and the tile in inches. Since the floor is 6ft x 3ft. we can say it is 72in x 36in, because 12 inches equals 1 foot. If the tiles are 9in x 9in we can fit 8 tiles along the length and 4 tiles along the width. To find the total number of tiles we multiply 8 x 4 = 32. Alternately we could find the area of the floor (72 x 36, and divide by the area of the tile 9 x 9)

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Question

A rectangle has a perimeter of 40 inches. It is 3 times as long as it is wide. What is the area of the rectangle in square inches?

Answer

The width of the rectangle is w, therefore the length is 3w. The perimeter, P, can then be described as P = w + w + 3w +3w

40 = 8w

w = 5

width = 5, length = 3w = 15

A = 5*15 = 75 square inches

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Question

Angela is carpeting a rectangular conference room that measures 20 feet by 30 feet. If carpet comes in rectangular pieces that measures 5 feet by 4 feet, how many carpet pieces will she need to carpet the entire room?

Answer

First, we need to find the area of the room. Because the room is rectangular, we can multiply 20 feet by 30 feet, which is 600 square feet. Next, we need to know how much space one carpet piece covers. Because the carpet pieces are also rectangular, we can multiply 4 feet by 5 feet to get 20 feet. To determine how many pieces of carpet Angela will need, we must divide the total square footage of the room (600 feet) by the square footage covered by one carpet piece (20 feet). 600 divided by 20 is 30, so Angela will need 30 carpet pieces to carpet the entire room.

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Question

If the area Rectangle A is larger than Rectangle B and the sides of Rectangle A are and , what is the area of Rectangle B?

Answer

$A = \text{length}: \times :\text{width} = 10 \times 6 = 60$

$B =\frac{60}{1.25} = 48$

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Question

If the width of a rectangle is 8 inches, and the length is half the width, what is the area of the rectangle in square inches?

Answer

the length of the rectangle is half the width, and the width is 8, so the length must be half of 8, which is 4.

The area of the rectangle can be determined from multiplying length by width, so,

4 x 8 = 32 inches squared

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Question

The front façade of a building is 100 feet tall and 40 feet wide. There are eight floors in the building, and each floor has four glass windows that are 8 feet wide and 6 feet tall along the front façade. What is the total area of the glass in the façade?

Answer

Glass Area per Window = 8 ft x 6 ft = 48 ft2

Total Number of Windows = Windows per Floor * Number of Floors = 4 * 8 = 32 windows

Total Area of Glass = Area per Window * Total Number of Windows = 48 * 32 = 1536 ft2

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Question

If Mrs. Stietz has a patio that measures 96 inches by 72 inches and she wants to cover it with stone tiles that measure one foot by half a foot, what is the minimum number of tiles she needs to cover the patio?

Answer

96. Converting the dimensions of the tiles to inches, they each measure 12 inches by 6 inches. This means that there need to be 8 tiles to span the length of the patio, and 6 tiles to span the width of the patio. She needs to cover the entire area, so we can multiply 8 times 12 to get 96, the number of tiles she needs for the patio.

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Question

The length of a rectangle is 5 times its width. Its width is 3 inches long. What is the area of the rectangle in square inches?

Answer

The length is 5 x 3 = 15 inches. Multiplied by the width of 3 inches, yields 45 in2.

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Question

The rectangular bathroom floor in Michael’s house is ten feet by twelve feet. He wants to purchase square tiles that are four inches long and four inches wide to cover the bathroom floor. If each square tile costs $2.50, how much money will Michael need to spend in order to purchase enough tiles to cover his entire bathroom floor?

Answer

The dimensions for the bathroom are given in feet, but the dimensions of the tiles are given in inches; therefore, we need to convert the dimensions of the bathroom from feet to inches, because we can’t compare measurements easily unless we are using the same type of units.

Because there are twelve inches in a foot, we need to multiply the number of feet by twelve to convert from feet to inches.

10 feet = 10 x 12 inches = 120 inches

12 feet = 12 x 12 inches = 144 inches

This means that the bathroom floor is 120 inches by 144 inches. The area of Michael’s bathroom is therefore 120 x 144 in2 = 17280 in2.

Now, we need to find the area of the tiles in square inches and calculate how many tiles it would take to cover 17280 in2.

Each tile is 4 in by 4 in, so the area of each tile is 4 x 4 in2, or 16 in2.

If there are 17280 in2 to be covered, and each tile is 16 in2, then the number of tiles we need is 17280 ÷ 16, which is 1080 tiles.

The question ultimately asks us for the cost of all these tiles; therefore, we need to multiply 1080 by 2.50, which is the price of each tile.

The total cost = 1080 x 2.50 dollars = 2700 dollars.

The answer is $2700.

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Question

Ron has a fixed length of wire that he uses to make a lot. On Monday, he uses the wire to make a rectangular lot. On Tuesday, he uses the same length of wire to form a square-shaped lot. Ron notices that the square lot has slightly more area, and he determines that the difference between the areas of the two lots is sixteen square units. What is the positive difference, in units, between the length and the width of the lot on Monday?

Answer

Let’s say that the rectangular lot on Monday has a length of l and a width of w. The area of a rectangular is the product of the length and the width, so we can write the area of the lot on Monday as lw.

Next, we need to find an expression for the area of the lot on Tuesday. We are told that the lot is in the shape of a square and that it uses the same length of wire. If the length of the wire used is the same on both days, then the perimeter will have to remain the same. In other words, the perimeter of the square will equal the perimeter of the rectangle. The perimeter of a rectangle is given by 2_l_ + 2_w_.

We also know that if s is the length of a side of a square, then the perimeter is 4_s_, because each side of the square is congruent. Let’s write an equation that sets the perimeter of the rectangle and the square equal.

2_l_ + 2_w_ = 4_s_

If we divide both sides by 4 and then simplify the expression, then we can write the length of the square as follows:

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Question

A rectangle has a width of 2_x_. If the length is five more than 150% of the width, what is the area of the rectangle?

Answer

Given that w = 2_x_ and l = 1.5_w_ + 5, a substitution will show that l = 1.5(2_x_) + 5 = 3_x_ + 5.

A = lw = (3_x_ + 5)(2_x_) = 6_x_2 + 10_x_

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Question

Rectangle ABCD is shown in the figure above. Points A and B lie on the graph of y = 64 – _x_2 , and points C and D lie on the graph of y = _x_2 – 36. Segments AD and BC are both parallel to the y-axis. The x-coordinates of points A and B are equal to –k and k, respectively. If the value of k changes from 2 to 4, by how much will the area of rectangle ABCD increase?

Answer

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Question

A rectangle’s base is twice its height. If the base is 8” long, what is the area of the rectangle?

Answer

Rectangle

B = 2_H_

B = 8”

H = B/2 = 8/2 = 4”

Area = B x H = 8” X 4” = 32 in2

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Question

Two circles of a radius of each sit inside a square with a side length of . If the circles do not overlap, what is the area outside of the circles, but within the square?

Answer

The area of a square = $\dpi{100} \small side^{2}$

The area of a circle is $\dpi{100} \small \pi r^{2}$

Area = Area of Square $\dpi{100} \small -$ 2(Area of Circle) =

$8^{2}-2\cdot \pi (2)^{2}$

$64-8\pi$

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Question

George wants to paint the walls in his room blue. The ceilings are 10 ft tall and a carpet 12 ft by 15 ft covers the floor. One gallon of paint covers 400 $ft^{2}$ and costs $40. One quart of paint covers 100 $ft^{2}$ and costs $15. How much money will he spend on the blue paint?

Answer

The area of the walls is given by $A = 2wh + 2lh = 2(12)(10) + 2(15)(10) = 540 ft^{2}$

One gallon of paint covers 400 $ft^{2}$ and the remaining 140 $ft^{2}$ would be covered by two quarts.

So one gallon and two quarts of paint would cost

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Question

Daisy gets new carpet for her rectangluar room. Her floor is $21\ ft \times 24\ ft$ . The carpet sells for $5 per square yard. How much did she spend on her carpet?

Answer

Since $3\ ft=1\ yd$ the room measurements are 7 yards by 8 yards. The area of the floor is thus 56 square yards. It would cost $5\cdot 56=$280$ .

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Question

The length of a rectangular rug is five more than twice its width. The perimeter of the rug is 40 ft. What is the area of the rug?

Answer

For a rectangle, $P=2w+2l$ and $A=lw$ where $w$ is the width and $l$ is the length.

Let $x=width$ and $2x+5=length$ .

So the equation to solve becomes $40=2x+2(2x+5)$ or $40=6x+10$ .

Thus $x=5\ ft$ and $2x+5=15\ ft$ , so the area is $75\ ft^{2}$ .

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