Question

practice 2
the pyramid of khufu in giza, egypt, was the world’s tallest free - standing structure for more than 3,500 years. its original height was about 144 meters. its base is approximately a square with a side length of 231 meters.
the diagram shows a cross section created by dilating the base using the top of the pyramid as the center of dilation. the cross - section is at a height of 96 meters.

a. what scale factor was used to create the cross - section?
type your answer in the box.

b. what are the dimensions of the cross - section?
type your answers in the boxes.

meter - by - meter

Explanation:

Part a

Step1: Find the height from the top

The original height of the pyramid is 144 m. The cross - section is at a height of 96 m from the base, so the height from the top of the pyramid to the cross - section is \(144 - 96=48\) m.

Step2: Calculate the scale factor

For a dilation centered at the top of the pyramid, the scale factor \(k\) of the dilation (for similar figures, since the cross - section and the base are similar squares) is the ratio of the height from the top to the cross - section to the total height from the top to the base. The total height from the top to the base is 144 m. So the scale factor \(k=\frac{48}{144}=\frac{1}{3}\)? Wait, no. Wait, actually, the height from the top to the cross - section is \(h_1 = 144 - 96=48\) m, and the height from the top to the base is \(h_2 = 144\) m. Wait, no, maybe I got it reversed. Wait, the cross - section is a dilation of the base with the center at the top. So the ratio of the heights (from the top) should be equal to the scale factor. The height from the top to the cross - section is \(144 - 96 = 48\) m, and the height from the top to the base is 144 m. So the scale factor \(k=\frac{48}{144}=\frac{1}{3}\)? Wait, no, that can't be. Wait, maybe the height from the top to the cross - section is \(h = 144-96 = 48\), and the height from the top to the base is 144. So the scale factor is \(\frac{48}{144}=\frac{1}{3}\)? Wait, no, let's think again. The cross - section is at a height of 96 m from the base, so the distance from the top to the cross - section is \(144 - 96=48\) m. The original pyramid has a height of 144 m (from top to base). So the scale factor for the dilation (since the two figures, the cross - section square and the base square, are similar) is the ratio of their corresponding linear dimensions, which is equal to the ratio of their heights from the top. So scale factor \(k=\frac{48}{144}=\frac{1}{3}\)? Wait, no, that would make the cross - section smaller. Wait, but if the cross - section is at 96 m from the base, the height from the top is \(144 - 96 = 48\), and the height from the top to the base is 144. So the scale factor is \(\frac{48}{144}=\frac{1}{3}\)? Wait, maybe I made a mistake. Wait, the height of the smaller pyramid (the part above the cross - section) is \(144 - 96=48\) m, and the height of the original pyramid is 144 m. So the scale factor between the smaller pyramid (above cross - section) and the original pyramid is \(\frac{48}{144}=\frac{1}{3}\). But the cross - section is the base of the smaller pyramid, so the scale factor from the original base to the cross - section is \(\frac{1}{3}\)? Wait, no, the original base has side length 231 m. Let's check part b. If the scale factor is \(\frac{1}{3}\), then the side length of the cross - section would be \(231\times\frac{1}{3}=77\) m. But let's re - calculate the scale factor. The height from the top to the cross - section is \(h_1=144 - 96 = 48\) m, and the height from the top to the base is \(h_2 = 144\) m. So the scale factor \(k=\frac{h_1}{h_2}=\frac{48}{144}=\frac{1}{3}\). Wait, that seems correct.

Step1: Determine the height from the top to the cross - section

The original height of the pyramid is \(H = 144\) m. The cross - section is at a height of \(h_{base}=96\) m from the base. So the height from the top of the pyramid to the cross - section is \(h = H - h_{base}=144 - 96 = 48\) m.

Step2: Calculate the scale factor

For similar figures (the cross - section square and the base square, since the dilation is centered at the top of the pyramid), the scale factor \(k\) is the ratio of the…

Step1: Recall the scale factor

From part (a), the scale factor \(k = \frac{1}{3}\).

Step2: Calculate the side length of the cross - section

The base of the pyramid is a square with side length \(s = 231\) m. Since the cross - section is a square (because the base is a square and the dilation is centered at the top, so the cross - section is also a square) and the scale factor of dilation is \(k=\frac{1}{3}\), the side length of the cross - section \(s'\) is given by \(s'=s\times k\). Substituting \(s = 231\) m and \(k=\frac{1}{3}\), we get \(s'=231\times\frac{1}{3}=77\) m. Since it's a square, both dimensions are 77 meters.

Answer:

\(\frac{1}{3}\)

Part b