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: 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?
image of a grid
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 \( h \) and the height of the stone portion be \( \frac{1}{2} \) foot, and the height of the brick portion be \( h_b \). First, find the total height of the wall.
Using \( A = l\times h \), we have \( h=\frac{A}{l} \). Substituting \( A = \frac{187}{8} \) and \( l=\frac{17}{2} \), we get \( h=\frac{\frac{187}{8}}{\frac{17}{2}}=\frac{187}{8}\times\frac{2}{17}=\frac{187\times2}{8\times17}=\frac{374}{136}=\frac{11}{4} = 2\frac{3}{4} \) feet.

Step2: Find the height of the brick portion

The total height \( h \) is the sum of the height of the stone portion (\( \frac{1}{2} \) foot) and the height of the brick portion (\( h_b \)). So, \( h_b=h - \frac{1}{2} \). Substituting \( h=\frac{11}{4} \) and \( \frac{1}{2}=\frac{2}{4} \), we get \( h_b=\frac{11}{4}-\frac{2}{4}=\frac{9}{4}=2\frac{1}{4} \) feet. Alternatively, we can also find the area of the stone portion first. The area of the stone portion \( A_s=l\times h_s \), where \( l = \frac{17}{2} \) and \( h_s=\frac{1}{2} \), so \( A_s=\frac{17}{2}\times\frac{1}{2}=\frac{17}{4} \) square feet. Then the area of the brick portion \( A_b=A - A_s=\frac{187}{8}-\frac{17}{4}=\frac{187}{8}-\frac{34}{8}=\frac{153}{8} \) square feet. Then, using \( A_b = l\times h_b \), we have \( h_b=\frac{A_b}{l}=\frac{\frac{153}{8}}{\frac{17}{2}}=\frac{153}{8}\times\frac{2}{17}=\frac{153\times2}{8\times17}=\frac{306}{136}=\frac{9}{4}=2\frac{1}{4} \) feet.

To find the fraction of the wall that is brick, we can find the ratio of the area of the brick portion to the total area of the wall, or the ratio of the height of the brick portion to the total height of the wall (since the length is the same for both portions, the ratio of areas is equal to the ratio of heights).
From Part A, the total height \( h=\frac{11}{4} \) feet and the height of the brick portion \( h_b=\frac{9}{4} \) feet? Wait, no, wait. Wait, the stone portion height is \( \frac{1}{2}=\frac{2}{4} \) feet, total height is \( \frac{11}{4} \) feet, so brick height is \( \frac{11}{4}-\frac{2}{4}=\frac{9}{4} \)? Wait, no, that can't be. Wait, let's recalculate the total height. Wait, area is \( 23\frac{3}{8}=\frac{187}{8} \), length is \( 8\frac{1}{2}=\frac{17}{2} \). So \( h=\frac{187/8}{17/2}=\frac{187}{8}\times\frac{2}{17}=\frac{187}{68}=\frac{11}{4} = 2.75 \) feet. Stone height is \( 0.5 \) feet, so brick height is \( 2.75 - 0.5 = 2.25=\frac{9}{4} \) feet? Wait, but let's check the area of brick. Area of brick: length \( \frac{17}{2} \), height \( \frac{9}{4} \), so area is \( \frac{17}{2}\times\frac{9}{4}=\frac{153}{8}=19.125 \) square feet. Total area is \( 23.375 \) square feet. Stone area: \( \frac{17}{2}\times\frac{1}{2}=\frac{17}{4}=4.25 \) square feet. \( 19.125 + 4.25 = 23.375 \), which matches. Now, the fraction of brick is \( \frac{\text{Area of brick}}{\text{Total area}}=\frac{\frac{153}{8}}{\frac{187}{8}}=\frac{153}{187}=\frac{9}{11} \)? Wait, no, \( 153\div17 = 9 \), \( 187\div17 = 11 \), so \( \frac{153}{187}=\frac{9}{11} \)? Wait, but let's check the height ratio. Brick height \( \frac{9}{4} \), total height \( \frac{11}{4} \), so ratio is \( \frac{9/4}{11/4}=\frac{9}{11} \). Wait, but the options are A \( \frac{9}{11} \), B \( \frac{2}{11} \), C \( \frac{2}{3} \), D \( \frac{3}{8} \). Wait, but maybe I made a mistake in total height. Wait, let's recalculate total height. \( A = l\times h \), so \( h=\frac{A}{l} \). \( A = 23\frac{3}{8}=\frac{187}{8} \), \( l = 8\frac{1}{2}=\frac{17}{2} \). So \( h=\frac{187}{8}\div\frac{17}{2}=\frac{187}{8}\times\frac{2}{17}=\frac{187\times2}{8\times17}=\frac{374}{136}=\frac{187}{68}=\frac{11}{4} \)? Wait, \( 17\times11 = 187 \), \( 8\times2 = 16 \)? No, wait, \( 8\times17 = 136 \), \( 187\times2 = 374 \), \( 374\div136 = 2.75=\frac{11}{4} \), that's correct. Stone height is \( \frac{1}{2} \) foot, so brick height is \( \frac{11}{4}-\frac{1}{2}=\frac{11}{4}-\frac{2}{4}=\frac{9}{4} \) foot. Then the fraction of brick is \( \frac{\text{Brick height}}{\text{Total height}}=\frac{\frac{9}{4}}{\frac{11}{4}}=\frac{9}{11} \), which is option A.

Answer:

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

Part B