Use the formula for the surface area of a square pyramid to solve for the slant height if the length of the base is 40 in. and SA equals 2,800 in2
The formula for the surface area of a square pyramid is given by:
SA = B + (1/2)Pl
Where:
B = area of the base
P = perimeter of the base
l = slant height
In the given problem, the length of the base is 40 in., and the surface area is 2,800 in².
We need to find the slant height, so let's start by finding the area of the base (B).
The base is a square, so we can use the formula for the area of a square:
Area of a square = side * side
The side of the square base can be found by using the formula for the perimeter of a square:
Perimeter of a square = 4 * side
Since the length of the base is given as 40 in., each side of the base is 40 in. / 4 = 10 in.
Now, we can find the area of the base:
B = 10 in. * 10 in. = 100 in²
Substituting the values of B, P, and SA into the formula, we have:
2,800 in² = 100 in² + (1/2)(40 in.)(P)
To find the value of P, we need to find the perimeter of the base. Since it is a square, all sides are equal. So, the perimeter is:
P = 4 * side = 4 * 10 in. = 40 in.
Now, let's substitute the value of P into the equation:
2,800 in² = 100 in² + (1/2)(40 in.)(40 in.)
Simplifying further:
2,800 in² = 100 in² + (1/2)(1600 in²)
2,800 in² = 100 in² + 800 in²
2,800 in² = 900 in²
Now, let's isolate the slant height term:
(1/2)(1600 in²) = 2,800 in² - 100 in²
800 in² = 2,700 in²
To solve for the slant height (l), divide both sides of the equation by (1/2)(1600 in²):
l = 2,700 in² / 800 in²
l = 3.375 in.
Therefore, the slant height of the square pyramid is approximately 3.375 inches.