Exercise 25 Page 860

To find the slant height of the given pyramid, we will first find the base area of the pyramid. Then we will use the formula for the surface area to calculate the slant height.

Base Area

The base of the pyramid is a right triangle with a width of $24\text{ cm}$ and a hypotenuse of $25\text{ cm}.$ We can find its area using the known formula where $b$ is a base and $h$ is its corresponding height. Note that the width of the triangle is the measure of one of the legs. Let's now find the measure of the second leg of the triangle by using the Pythagorean Theorem. When doing this, recall that we are given the value of the hypotenuse. We only kept the principal root when solving the equation because $s$ is the measure of the leg of a triangle and it must be non-negative. Therefore, the measure of the second leg is $7\text{ cm}.$ Since the triangle is a right triangle, this leg is the height corresponding to the other leg. Let's now substitute $b=24$ and $h=7$ into the formula for the area of the triangle. We have found that the area of the base of the pyramid is $A=84{\text{ cm}}^2.$

Slant Height

To calculate the slant height of the pyramid, we can use the formula for the surface area where $P$ is the perimeter of the base, $\ell$ is the slant height, and $A$ is the area of the base. Let's first find the perimeter of the base of the pyramid. To do so we can add all of the three side lengths of the triangle. We will now substitute the values of the $\text{surface area},$ the $\text{base area},$ and the $\text{perimeter}$ into the formula for the surface area to find the slant height. The slant height of the pyramid is $16\text{ cm}.$