Question

type the correct answer in the box. use numerals instead of words. a right square pyramid has an altitude of 10 and each side of the base is 6. to the nearest tenth of a centimeter, what is the distance from the apex, or top of the pyramid, to each vertex of the base? a right square pyramid diagram is shown with altitude 10, base side 6, and the distance from apex to base vertex labeled x.

Explanation:

Step1: Find the distance from the center of the base to a vertex

The base is a square with side length \(s = 6\). The distance from the center of a square to a vertex \(d\) can be found using the Pythagorean theorem. For a right - triangle formed by half of the sides of the square. If we consider half of the side length \(a=\frac{6}{2}=3\), and for a square, the distance from the center to a vertex \(d\) (using the Pythagorean theorem for a right - triangle with two legs of length \(a = 3\)): \(d=\sqrt{3^{2}+3^{2}}=\sqrt{9 + 9}=\sqrt{18}=3\sqrt{2}\).

Step2: Use the Pythagorean theorem to find the distance from the apex to a vertex

Let the altitude of the pyramid \(h = 10\), and the distance from the center of the base to a vertex \(d = 3\sqrt{2}\). The distance \(x\) from the apex to a vertex of the base forms a right - triangle with legs \(h = 10\) and \(d=3\sqrt{2}\). By the Pythagorean theorem \(x=\sqrt{10^{2}+(3\sqrt{2})^{2}}\).
First, calculate \((3\sqrt{2})^{2}=9\times2 = 18\) and \(10^{2}=100\). Then \(x=\sqrt{100 + 18}=\sqrt{118}\).
Now, \(\sqrt{118}\approx10.9\)

Answer:

\(10.9\)