PSAT Math - How to find the area of a rectangle

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?

Explanation

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

Para-rec1

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?

272

544

176

352

Explanation

Para-rec2

Para-rec3

Rectangle

Note: Figure NOT drawn to scale

Give the ratio of the perimeter of Rectangle to that of Rectangle .

Explanation

The perimeter of Rectangle is

Opposite sides of a rectangle are congruent, so

and

The perimeter of Rectangle is

Opposite sides of a rectangle are congruent, so

and

The ratio of the perimeters is

$\frac{84}{60} = \frac{84 \div 12}{60 \div 12} = \frac{7}{5}$ - that is, 7 to 5.

Rectangle

Note: Figure NOT drawn to scale

What percent of Rectangle is pink?

$48\frac{3}{4} %$

$55 \frac{5}{9} %$

$44 \frac{4}{9} %$

$46\frac{2}{3} %$

Explanation

The pink region is Rectangle . Its length and width are

so its area is the product of these, or

$A = 13 \times 9 = 117$ .

The length and width of Rectangle are

so its area is the product of these, or

$20 \times 12 = 240$ .

We want to know what percent 117 is of 240, which can be answered as follows:

$\frac{117}{240} \times 100 % = 48\frac{3}{4} %$

Rectangle

Note: Figure NOT drawn to scale

Refer to the above diagram.

40% of Rectangle is pink. .

Evaluate .

Explanation

Rectangle has length and width , so it has area

$LW = 20 \times 15 = 300$ .

300 is 40% of, or 0.40 times, the area of Rectangle , which we will call . We can determine as follows:

$A = 300 \div 0.4 = 750$ .

The length of Rectangle is

so its width is

$W = AF = 750 \div 30 = 25$ .

Since

Rectangle

Note: Figure NOT drawn to scale

What percent of Rectangle is white?

$33 \frac{1}{3} %$

$44 \frac{4}{9} %$

Explanation

The pink region is Rectangle . Its length and width are

so its area is the product of these, or

$24 \times 12 = 288$ .

The length and width of Rectangle are

so its area is the product of these, or

$30 \times 15 = 450$ .

The white region is Rectangle cut from Rectangle , so its area is the difference of the two:

So we want to know what percent 162 is of 450, which can be answered as follows:

$\frac{162}{450} \times 100 % = 36 %$

Garden

Refer to the above figure, which shows a rectangular garden (in green) surrounded by a dirt path (in orange). The dirt path is seven feet wide throughout. Which of the following polynomials gives the area of the dirt path in square feet?

Explanation

The area of the dirt path is the difference between the areas of the outer and inner rectangles.

The outer rectangle has area

$A = 2x \cdot x = 2x^{2}$

The area of the inner rectangle can be found as follows:

The length of the garden is $2 \times 7= 14$ feet less than that of the entire lot, or

;

The width of the garden is $2 \times 7= 14$ less than that of the entire lot, or

;

The area of the garden is their product:

$=2x \cdot x - 2x \cdot 14 - 14 \cdot x + 14 \cdot 14$

$= 2x ^{2} - 28x - 14 x + 196$

$= 2x ^{2} - 42 x + 196$

Now, subtract the areas:

$2x ^{2} - \left (2x ^{2} - 42 x + 196 \right ) = 2x ^{2} - 2x ^{2} + 42 x - 196 = 42x - 196$

Garden

Note: Figure NOT drawn to scale

Refer to the above figure, which shows a rectangular garden (in green) surrounded by a dirt path (in orange) eight feet wide throughout. What is the area of that dirt path?

$2,304\textrm{ ft}^{2}$

$1,216\textrm{ ft}^{2}$

$2,560\textrm{ ft}^{2}$

$1,280\textrm{ ft}^{2}$

The correct area is not given among the other responses.

Explanation

The dirt path can be seen as the region between two rectangles. The outer rectangle has length and width 100 feet and 60 feet, respectively, so its area is

$A = 100 \cdot 60 = 6,000$ square feet.

The inner rectangle has length and width $(100 - 2\times 8) = 84$ feet and $(60 - 2\times 8) = 44$ feet, respectively, so its area is

$A = 84 \times 44= 3,696$ square feet.

The area of the path is the difference of the two:

square feet.

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?