One of Pyramids
A soundproofing tile is made of 8 identical solid right pyramids with square bases. The length of the whole tile shown is [tex]$x$[/tex] inches.
Which expression shows the area of the base of each pyramid?
[tex]$x^2 \text { in. }{ }^2$[/tex]
[tex]$\left(\frac{1}{4} x\right)^2 \text{ in. }^2$[/tex]
[tex]$\left(\frac{1}{3} x\right)^2 \text{ in. }^2$[/tex]
[tex]$\left(\frac{1}{2} x\right)^2 \text{ in. }^2$[/tex]
Answer (2)
The problem involves finding the area of the base of a pyramid given the total length of a tile made of 8 identical pyramids.
We assume the pyramids are arranged in a 2x4 grid, so 4 pyramids fit along the length x .
We express the side length s of each pyramid's base as s = 4 x .
The area of each pyramid's base is then calculated as ( 4 1 x ) 2 .
Explanation
Problem Analysis Let's analyze the problem. We have a soundproofing tile made of 8 identical pyramids with square bases. The length of the tile is x inches. We need to find the area of the base of each pyramid.
Setting up the equation Assume the 8 pyramids are arranged in a 2x4 grid. This means there are 4 pyramids along the length x . Let s be the side length of the square base of each pyramid. Then, we can write the equation:
4 s = x
Solving for side length Now, let's solve for s in terms of x :
s = 4 x
Calculating the area The area of the square base of each pyramid is s 2 . Substitute s = 4 x into the area formula:
Area = s 2 = ( 4 x ) 2
Final Answer Therefore, the area of the base of each pyramid is ( 4 1 x ) 2 square inches.
Examples
Imagine you're designing a mosaic tile pattern using identical square tiles. If you know the total length the tiles will cover and how many tiles fit along that length, you can calculate the area of each individual tile. This is similar to our pyramid problem, where knowing the total length x and the arrangement of the pyramids allows us to find the area of each pyramid's base. This concept is useful in various design and construction scenarios where you need to determine individual component sizes based on overall dimensions.
The area of the base of each pyramid is given by the expression ( 4 1 x ) 2 square inches. This is derived from the arrangement of pyramids in a 2x4 grid, leading to the calculation of the side length of each pyramid's base. Thus, the correct answer is ( 4 1 x ) 2 in. 2 .
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