Question

luis has a pyramid - shaped plant pot. it has a square base with a side length of 36 cm, and the height of the pot is 38 cm.
luis wants to fill the pot with soil so that the soil takes up 75% of the pot’s volume.
how far up the pot will the soil reach?
round to the nearest tenth.
\boxed{} cm

Explanation:

Step1: Recall the volume formula for a pyramid

The volume \( V \) of a pyramid is given by \( V=\frac{1}{3}Bh \), where \( B \) is the area of the base and \( h \) is the height. For a square - based pyramid with side length \( s \) of the base, \( B = s^{2} \). So the volume of the whole pot (with height \( H = 38\) cm and base side length \( S=36\) cm) is \( V_{total}=\frac{1}{3}S^{2}H \).

Let the height of the soil - filled part be \( h \) and the side length of the base of the soil - filled (similar) pyramid be \( s \). Since the two pyramids (the whole pot and the soil - filled part) are similar, the ratio of their corresponding linear dimensions (side lengths and heights) is the same. That is, \( \frac{s}{S}=\frac{h}{H} \), so \( s=\frac{S}{H}h \).

The volume of the soil - filled part \( V_{soil}=\frac{1}{3}s^{2}h=\frac{1}{3}(\frac{S}{H}h)^{2}h=\frac{1}{3}\frac{S^{2}}{H^{2}}h^{3} \).

We know that \( V_{soil}=0.75V_{total} \). Substitute \( V_{total}=\frac{1}{3}S^{2}H \) and \( V_{soil}=\frac{1}{3}\frac{S^{2}}{H^{2}}h^{3} \) into the equation:

\( \frac{1}{3}\frac{S^{2}}{H^{2}}h^{3}=0.75\times\frac{1}{3}S^{2}H \)

We can cancel out \( \frac{1}{3}S^{2} \) from both sides of the equation (since \( S
eq0 \) and \( \frac{1}{3}
eq0 \)):

\( \frac{h^{3}}{H^{2}} = 0.75H \)

\( h^{3}=0.75H^{3} \)

Step2: Solve for \( h \)

We know that \( H = 38\) cm. First, calculate \( H^{3}=38^{3}=38\times38\times38 = 38\times1444=54872 \)

Then, \( 0.75H^{3}=0.75\times54872 = 41154 \)

Now, we need to find \( h \) such that \( h^{3}=41154 \). Take the cube - root of both sides:

\( h=\sqrt[3]{41154}\approx34.5 \) (We can also use the property of similar solids: For similar solids, the ratio of volumes is the cube of the ratio of corresponding linear measurements. Let the scale factor (ratio of heights) be \( k=\frac{h}{H} \). Then the ratio of volumes \( \frac{V_{soil}}{V_{total}}=k^{3} \). Since \( \frac{V_{soil}}{V_{total}} = 0.75 \), then \( k=\sqrt[3]{0.75} \). And \( h = kH=\sqrt[3]{0.75}\times38 \).

Calculate \( \sqrt[3]{0.75}\approx0.90856 \), then \( h = 0.90856\times38\approx34.5 \))

Answer:

\( 34.5 \)