Question

Look at the square pyramid below.



24 mi24\ \text{mi}24 mi lll

7 mi7\ \text{mi}7 mi

How many miles long is the slant height of the square pyramid? Click Here for Help Video.

Answer 1

wrong

Answer 2

To find the slant height of the square pyramid, we need to use the Pythagorean theorem.

We are given:

- The base length of the square pyramid \( b = 24 \, \text{mi} \)
- The vertical height of the pyramid \( h = 7 \, \text{mi} \)

First, we will find the distance from the center of the base to one of the corners of the base. This distance is half of the base length (since it's a square pyramid):

\[
\text{Half of the base length} = \frac{b}{2} = \frac{24 \, \text{mi}}{2} = 12 \, \text{mi}
\]

Next, we will create a right triangle with:

- One leg being the vertical height \( h = 7 \, \text{mi} \)
- The other leg being the distance from the center to the corner of the base \( 12 \, \text{mi} \)
- The hypotenuse being the slant height \( l \)

Using the Pythagorean theorem:

\[
l^2 = h^2 + \left(\frac{b}{2}\right)^2
\]

Substituting the known values:

\[
l^2 = (7 \, \text{mi})^2 + (12 \, \text{mi})^2
\]
\[
l^2 = 49 + 144
\]
\[
l^2 = 193
\]

Now, take the square root to find the slant height:

\[
l = \sqrt{193}
\]
\[
l \approx 13.89 \, \text{mi}
\]

Thus, the slant height of the square pyramid is approximately **13.89 miles**.

Answer 3

Let's re-evaluate the problem carefully.

We have:

- The pyramid has a square base with each side measuring \(24 \, \text{mi}\).
- The vertical height from the base to the apex of the pyramid is \(7 \, \text{mi}\).

**Step 1: Find the distance from the center of the base to a corner.**

For a square with side length \(s\), the distance from the center to a corner can be calculated using:

\[
\text{Distance} = \frac{s}{2}\sqrt{2}
\]

In this case, \(s = 24 \, \text{mi}\):

\[
\text{Distance from center to corner} = \frac{24 \, \text{mi}}{2} \sqrt{2} = 12 \, \text{mi} \sqrt{2}
\]

**Step 2: Use the Pythagorean theorem to find the slant height.**

The slant height \(l\) of the pyramid forms a right triangle with:

- One leg being the vertical height \(h = 7 \, \text{mi}\)
- The other leg being the distance we calculated above \(12 \, \text{mi} \sqrt{2}\)

Using the Pythagorean theorem:

\[
l^2 = h^2 + \left(12 \, \text{mi} \sqrt{2}\right)^2
\]

Calculating:

\[
l^2 = (7 \, \text{mi})^2 + (12 \, \text{mi} \sqrt{2})^2
\]

\[
l^2 = 49 + 144 \cdot 2
\]
\[
l^2 = 49 + 288
\]
\[
l^2 = 337
\]

Now, take the square root to find the slant height:

\[
l = \sqrt{337}
\]

So, the slant height of the square pyramid is:

\[
l \approx 18.36 \, \text{mi}
\]

I apologize for the earlier miscalculation. The correct slant height is approximately **18.36 miles**.