for the square pyramid, the lateral Edge length, e, is 25 in., and the slant height, s, is 24 in. find the height of the pyramid, h.
round the answer to the nearest whole number. hint: before finding the length of H, you will first need to find the length of r.
To find the height (h) of the square pyramid, we first need to determine the length of the base diagonal (d) and then the length of the apothem of the base (r), which is half the base diagonal. The slant height (s) is the hypotenuse of a right triangle formed by the height of the pyramid, h, and half the base diagonal, r.
Given:
- The lateral edge length (e) is 25 inches.
- The slant height (s) is 24 inches.
We can use the Pythagorean theorem to find r. Since the pyramid is square, all four edges of the base are equal in length, let's call it B.
First, we find B from e and s:
\[ \text{e}^2 = \text{B}^2 + \text{s}^2 \]
We know e and s, so we plug them in:
\[ 25^2 = \text{B}^2 + 24^2 \]
\[ \text{B}^2 = 25^2 - 24^2 \]
\[ \text{B}^2 = 625 - 576 \]
\[ \text{B}^2 = 49 \]
\[ \text{B} = \sqrt{49} \]
\[ \text{B} = 7 \, \text{inches} \]
Now, we have the edge of the base (B) is 7 inches. Next, we find the diagonal of the base (d):
\[ \text{d} = \text{B} \times \sqrt{2} \]
\[ \text{d} = 7 \times \sqrt{2} \]
Now, we find the apothem (r), which is half the diagonal of the base:
\[ \text{r} = \frac{\text{d}}{2} \]
\[ \text{r} = \frac{7 \times \sqrt{2}}{2} \]
\[ \text{r} = \frac{7}{2} \times \sqrt{2} \]
\[ \text{r} = 3.5 \times \sqrt{2} \]
\[ \text{r} \approx 3.5 \times 1.414 \]
\[ \text{r} \approx 4.949 \, \text{inches} \]
Now we have r, we can find the height (h) using the slant height (s) and r in a right-angled triangle:
\[ \text{s}^2 = \text{h}^2 + \text{r}^2 \]
So:
\[ 24^2 = \text{h}^2 + 4.949^2 \]
\[ \text{h}^2 = 576 - 4.949^2 \]
\[ \text{h}^2 = 576 - 24.49 \]
\[ \text{h}^2 = 551.51 \]
\[ \text{h} = \sqrt{551.51} \]
\[ \text{h} \approx 23.482 \]
Rounded to the nearest whole number:
\[ \text{h} \approx 23 \, \text{inches} \]
Therefore, the height of the pyramid is approximately 23 inches.