Question

1. employees of a landscaping company built a retaining wall with area ( 23\frac{3}{8} ) sq ft. they used stone for the lower portion of the wall and brick for the upper portion of the wall.

image of the wall: brick on top, stone at bottom, length ( 8\frac{1}{2} ) feet, stone height ( \frac{1}{2} ) foot
part a
what is the height of the brick portion of the wall, in feet?
bubble sheet image
part b
what fraction of the wall is brick?
a ( \frac{9}{11} )
b ( \frac{2}{11} )
c ( \frac{2}{3} )
d ( \frac{3}{8} )

Explanation:

Part A

Step1: Recall the area formula for a rectangle

The area \( A \) of a rectangle is given by \( A = \text{length} \times \text{height} \). Here, the length of the wall (both stone and brick portions) is \( 8\frac{1}{2}=\frac{17}{2} \) feet, and the total area of the wall is \( 23\frac{3}{8}=\frac{187}{8} \) square feet. Let the total height of the wall be \( h \). So we have \( \frac{187}{8}=\frac{17}{2}\times h \).

Step2: Solve for total height \( h \)

To find \( h \), we can rearrange the formula: \( h=\frac{\frac{187}{8}}{\frac{17}{2}} \). Dividing by a fraction is multiplying by its reciprocal, so \( h = \frac{187}{8}\times\frac{2}{17}=\frac{187\times2}{8\times17}=\frac{374}{136}=\frac{11}{4} = 2\frac{3}{4}\) feet? Wait, no, wait. Wait, the stone portion has a height of \( \frac{1}{2} \) foot. Wait, maybe I misread. Wait, the total area is \( 23\frac{3}{8} \), length is \( 8\frac{1}{2} \). Let's first find the total height of the wall. Using \( A = l\times h \), so \( h=\frac{A}{l} \). So \( A = 23\frac{3}{8}=\frac{23\times8 + 3}{8}=\frac{187}{8} \), \( l = 8\frac{1}{2}=\frac{17}{2} \). Then \( h=\frac{187}{8}\div\frac{17}{2}=\frac{187}{8}\times\frac{2}{17}=\frac{187\times2}{8\times17}=\frac{374}{136}=\frac{11}{4}=2\frac{3}{4} \) feet. Then the stone portion is \( \frac{1}{2} \) foot, so the brick portion height is total height minus stone height: \( 2\frac{3}{4}-\frac{1}{2}=2\frac{3}{4}-\frac{2}{4}=2\frac{1}{4}=\frac{9}{4} \)? Wait, that doesn't seem right. Wait, maybe the wall is a rectangle, so area is length times total height. Wait, the stone portion is a rectangle with length \( 8\frac{1}{2} \) and height \( \frac{1}{2} \), so area of stone is \( 8\frac{1}{2}\times\frac{1}{2}=\frac{17}{2}\times\frac{1}{2}=\frac{17}{4} \) square feet. Then the area of brick is total area minus stone area: \( 23\frac{3}{8}-\frac{17}{4}=\frac{187}{8}-\frac{34}{8}=\frac{153}{8} \) square feet. Then the height of brick is area of brick divided by length: \( \frac{153}{8}\div\frac{17}{2}=\frac{153}{8}\times\frac{2}{17}=\frac{153\times2}{8\times17}=\frac{306}{136}=\frac{9}{4}=2\frac{1}{4} \) feet. Wait, but maybe there's a simpler way. Wait, total height \( h \) is \( \frac{A}{l}=\frac{23\frac{3}{8}}{8\frac{1}{2}}=\frac{\frac{187}{8}}{\frac{17}{2}}=\frac{187}{8}\times\frac{2}{17}=\frac{11}{4}=2.75 \) feet. Stone height is \( 0.5 \) feet, so brick height is \( 2.75 - 0.5 = 2.25=\frac{9}{4}=2\frac{1}{4} \) feet. Wait, but the answer options in the bubble sheet—wait, maybe I made a mistake. Wait, let's check again. Total area: \( 23\frac{3}{8}=\frac{187}{8} \), length \( 8\frac{1}{2}=\frac{17}{2} \). So total height \( h=\frac{187}{8}\div\frac{17}{2}=\frac{187\times2}{8\times17}=\frac{374}{136}=\frac{11}{4}=2\frac{3}{4} \) feet. Stone height is \( \frac{1}{2} \) foot, so brick height is \( 2\frac{3}{4}-\frac{1}{2}=2\frac{1}{4} \) feet, which is \( \frac{9}{4} \) or \( 2.25 \). Wait, but maybe the problem is that the wall is a rectangle, so the total height is area divided by length. Then brick height is total height minus stone height. So that's the way.

Wait, maybe the initial approach was wrong. Let's start over. The wall is a rectangle, so area = length × total height. So total height \( H = \frac{\text{Area}}{\text{Length}} \). Area is \( 23\frac{3}{8} = \frac{187}{8} \) sq ft, length is \( 8\frac{1}{2} = \frac{17}{2} \) ft. So \( H = \frac{187}{8} \div \frac{17}{2} = \frac{187}{8} \times \frac{2}{17} = \frac{187 \times 2}{8 \times 17} = \frac{374}{136} = \frac{11}{4} = 2\frac{3}{4} \) ft. The stone portion has height \( \frac{1}{2} \) ft, s…

Step1: Find the height of brick (from Part A)

From Part A, the height of the brick portion is \( \frac{9}{4} \) feet, and the height of the stone portion is \( \frac{1}{2} \) feet. So the total height of the wall is \( \frac{9}{4}+\frac{1}{2}=\frac{9}{4}+\frac{2}{4}=\frac{11}{4} \) feet.

Step2: Calculate the fraction of brick

The fraction of the wall that is brick is the height of brick divided by total height: \( \frac{\frac{9}{4}}{\frac{11}{4}}=\frac{9}{4}\times\frac{4}{11}=\frac{9}{11} \). Wait, but let's check with areas. Area of brick is \( \frac{153}{8} \) (from Part A: total area \( \frac{187}{8} \) minus stone area \( \frac{17}{4}=\frac{34}{8} \), so \( \frac{187 - 34}{8}=\frac{153}{8} \)). Total area is \( \frac{187}{8} \). So fraction is \( \frac{\frac{153}{8}}{\frac{187}{8}}=\frac{153}{187}=\frac{9}{11} \) (since 153 ÷ 17 = 9, 187 ÷ 17 = 11). So the fraction is \( \frac{9}{11} \), which is option A.

Answer:

\( \frac{9}{4} \) (or \( 2\frac{1}{4} \))

Part B