Rectangles - ISEE Upper Level Mathematics Achievement

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Question

A rectangle and a square have the same perimeter. The area of the square is square centimeters; the length of the rectangle is centimeters. Give the width of the rectangle in centimeters.

Answer

The sidelength of a square with area square centimeters is $\sqrt{225} = 15$ centimeters; its perimeter, as well as that of the rectangle, is therefore $15 \times 4 = 60$ centimeters.

Using the formula for the perimeter of a rectangle, substitute and solve for as follows:

$2 \cdot 21 + 2W = 60$

$2W \div 2 = 18\div 2$

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Question

A rectangle on the coordinate plane has its vertices at the points .

What percent of the rectangle is located in Quadrant I?

Answer

The total area of the rectangle is

$\left [ 3- (-6) \right ] \times\left [ 4- (-2) \right ] = 9 \times 6 = 54$ .

The area of the portion of the rectangle in Quadrant I is

$\left ( 3- 0 \right ) \times\left ( 4- 0 \right) = 3 \times 4 = 12$ .

Therefore, the portion of the rectangle in Quadrant I is

$\frac{12}{54} = \frac{2}{9} = \frac{2}{9} \times 100 % = 22 \frac{2}{9} %$ .

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Question

In a rectangle, the width is while the length is . If , what is the area of the rectangle?

Answer

In a rectangle in which the width is x while the length is 4x, the first step is to solve for x. If , the value of x can be found by dividing each side of this equation by 3.

Doing so gives us the information that x is equal to 3.

Thus, the area is equal to:

$Area=x\cdot4x$

$Area=3\cdot4\cdot3$

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Question

Your geometry book has a rectangular front cover which is 12 inches by 8 inches.

What is the area of your book cover?

Answer

Your geometry book has a rectangular front cover which is 12 inches by 8 inches.

What is the area of your book cover?

To find the area of a rectangle, use the following formula:

$A_{rectangle}=l*w$

Plug in our knowns and solve:

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Question

Find the area of a rectangle with a width of 5in and a length that is three times the width.

Answer

To find the area of a rectangle, we will use the following formula:

$\text{area of rectangle} = l \cdot w$

where l is the length and w is the width of the rectangle.

Now, we know the width of the rectangle is 5in. We also know the length is three times the width. Therefore, the length is 15in.

Knowing this, we can substitute into the formula. We get

$\text{area of rectangle} = 15\text{in} \cdot 5\text{in}$

$\text{area of rectangle} = 75\text{in}^2$

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Question

Find the area of a rectangle with a width of 7cm and a length that is four times the width.

Answer

To find the area of a rectangle, we will use the following formula:

$A = l \cdot w$

where l is the length and w is the width of the rectangle.

Now, we know the width of the rectangle is 7cm. We also know the length is four times the width. Therefore, the length is 28cm.

Knowing this, we will substitute into the formula. We get

$A = 28\text{cm} \cdot 7\text{cm}$

$A = 196\text{cm}^2$

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Question

Find the area of a rectangle with a width of 8cm and a length that is four times the width.

Answer

To find the area of a rectangle, we will use the following formula:

$A = l \cdot w$

where l is the length and w is the width of the rectangle.

Now, we know the width of the rectangle is 8cm. We also know the length of the rectangle is four times the width. Therefore, the length of the rectangle is 32cm.

Knowing this, we will substitute into the formula. We get

$A = 32\text{cm} \cdot 8\text{cm}$

$A = 256\text{cm}^2$

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Question

You are designing a poster to put on the front of your refrigerator. If the refrigerator door is 2 feet wide by 4.5 feet tall, what is the area of the largest poster you could fit on the door?

Answer

You are designing a poster to put on the front of your refrigerator. If the refrigerator door is 2 feet wide by 4.5 feet tall, what is the area of the largest poster you could fit on the door?

We need to find the area of a shape. Given the context of a refrigerator door and a poster, we can assume that the poster will be a rectangle. To find the area of a rectangle, we need to multiply length and width.

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Question

Find the area of a rectangle with a length of 12in and a width that is a third of the length.

Answer

To find the area of a rectangle, we will use the following formula:

$A = l \cdot w$

where l is the length and w is the width of the rectangle.

Now, we know the length of the rectangle is 12in. We also know the width of the rectangle is a third of the length. Therefore, the width is 4in.

So, we get

$A = 12\text{in} \cdot 4\text{in}$

$A = 48\text{in}^2$

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Question

A rectangle has a width of 9cm and a length that is three times the width. Find the area.

Answer

To find the area of a rectangle, we will use the following formula:

$A = l \cdot w$

where l is the length and w is the width of the rectangle.

Now, we know the width of the rectangle is 9cm. We also know the length of the rectangle is three times the width. Therefore, the length is 27cm. So, we can substitute.

$A = 27\text{cm} \cdot 9\text{cm}$

$A = 243\text{cm}^2$

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Question

The area of a rectangle is 100 square inches. What is the width of the rectangle?

Answer

Without more information we cannot determine the width of a rectangle based on its area alone.

If we know this is a square and therefore that all sides are equal, we could simply take the square root of 100, which is 10.

However, we do not know that all sides are equal. We need to know the length of the rectangle to solve this problem. Therefore the best we can do is see what the equation would look like if we knew the length.

$w\times l=100$

or

$w=\frac{100}{l}$

where is the width and is the length.

Note: the incorrect answer choices are all factors of 100 and possible answers, but we can't be sure which, if any, are the right one.

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Question

A rectangle has perimeter 56. Its length is ; give its width in terms of .

Answer

Substitute in the formula for the perimeter of a rectangle:

$2 \left ( 3x-8 \right ) + 2W = 56$

$2\cdot 3x-2\cdot 8 \right ) + 2W = 56$

$2W \div 2 =\left ( -6x + 72 \right ) \div 2$

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Question

The area of a rectangle is 40; its width is . Which of the following expressions is equal to the length of the rectangle?

Answer

Let be the length. The area of the rectangle is . Replace and solve for :

$L \left ( s+5 \right ) = 40$

$L \left ( s+5 \right ) \div \left ( s+5 \right ) = 40 \div \left ( s+5 \right )$

$L = \frac{40}{s + 5}$

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Question

The perimeter of a rectangle is 70; its width is . Which of the following expressions is equal to the length of the rectangle?

Answer

Let be the length. The perimeter of the rectangle is .

Replace and solve for :

$2L + 2 \left ( t+16 \right ) = 70$

$2L \div 2 =\left ( 38 -2t \right ) \div 2$

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Question

You have a shed that you use for storing your gardening supplies and outdoor tools. If the shed has a footprint of and a width of 16 ft, what is the length of the shed? (assume the base is a rectangle)

Answer

You have a shed that you use for storing your gardening supplies and outdoor tools. If the shed has a footprint of and a width of 16 ft, what is the length of the shed? (assume the base is a rectangle)

We are given a rectangle with a known area and width, and asked to find the length.

To do so, recall this formula:

We know A and w, so simply work backwards to solve for l

$l=\frac{A}{w}=\frac{284ft^2}{16ft}=17.75ft$

So, our answer is 17.75 ft

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Question

. Which is the greater quantity?

(a) The perimeter of the square on the coordinate plane with vertices

(b) The perimeter of the rectangle on the coordinate plane with vertices

Answer

(a) The square has sidelength , and therefore has perimeter .

(b) The rectangle has width and height , so therefore has perimeter

$\left ( A + 2 \right ) + \left ( A - 2 \right ) + \left ( A + 2 \right ) + \left ( A - 2\right ) = A + A + A + A + 2- 2 + 2 - 2 = 4A$ .

The perimeters are the same.

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Question

Your geometry book has a rectangular front cover which is 12 inches by 8 inches.

What is the perimeter of your book cover?

Answer

Your geometry book has a rectangular front cover which is 12 inches by 8 inches.

What is the perimeter of your book cover?

Perimeter is the distance around the outside of a shape.

Perimeter of a rectangle can be found via the following:

$P_{rectangle}=2l+2w$

Plug in our length and width to find our answer:

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Question

Your aunt and uncle live on a rectangular plot of land out in the countryside. The width of their plot is $\frac{3}{4}$ of a kilometer long. The length is twice the width. What is the total length of the perimeter of the plot of land?

Answer

Your aunt and uncle live on a rectangular plot of land out in the countryside. The width of their plot is $\frac{3}{4}$ of a kilometer long. The length is twice the width. What is the total length of the perimeter of the plot of land?

Begin by finding both the width and length.

We are told that the width is $\frac{3}{4}$ of a kilometer

Further, we are told that the length is twice the width. Write this as an equation and solve:

$l=2w=2\frac{3}{4}=\frac{6}{4}=1\frac{1}{2}$

Now, find the perimeter via the following:

$P=2l+2w\rightarrow 2(\frac{3}{4}km)+ 2(\frac{3}{2}km)=1\frac{1}{2}km+3km=4.5km$

So, our perimeter is 4.5 km

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