A soundproofing tile is made of 8 identical solid right pyramids with square bases. The length of the whole tile shown is $x$ inches. Which expression shows the area of the base of each pyramid?
$\left(\frac{1}{4} x\right)^2$ in. $^2$
$\left(\frac{1}{3} x\right)^2$ in. ${ }^2$
$\left(\frac{1}{2} x\right)^2$ in. ${ }^2$
$x^2$ in. $^2$
Answer (2)
Determine that the side length of each pyramid's base is 4 x .
Calculate the area of the square base by squaring the side length: ( 4 x ) 2 .
Simplify the expression to find the area of the base of each pyramid.
The area of the base of each pyramid is ( 4 1 x ) 2 square inches.
Explanation
Problem Analysis The soundproofing tile is made up of 8 identical right pyramids with square bases. The length of the entire tile is given as x inches. We need to find the area of the base of each pyramid. Since the tile is made of 8 identical pyramids arranged in 2 rows of 4 pyramids each, the length x corresponds to 4 pyramids lined up along that length.
Find the side length of the base To find the side length of the square base of each pyramid, we divide the total length x by the number of pyramids along that length, which is 4. Therefore, the side length of the base of each pyramid is 4 x inches.
Calculate the area of the base Now, to find the area of the square base, we square the side length: A re a = ( 4 x ) 2
Final Answer Therefore, the area of the base of each pyramid is ( 4 1 x ) 2 square inches.
Examples
Understanding area calculations is crucial in various real-world applications. For instance, when designing a room, knowing the area helps in determining the amount of flooring needed. Similarly, in gardening, calculating the area of a plot helps in estimating the amount of soil and plants required. In construction, calculating the area of surfaces is essential for estimating the amount of paint or cladding needed. These calculations ensure efficient use of materials and resources, saving time and money.
The area of the base of each pyramid is calculated to be ( 4 1 x ) 2 square inches. This is found by determining that the side length of each pyramid's base is 4 x inches and squaring it. Therefore, the correct answer is ( 4 1 x ) 2 in.².
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