Question

the total surface area of this square pyramid is 144 square feet
x ft.
6 ft. 6 ft.
what is the value of x?
4 feet
6 feet
9 feet
10 feet

Explanation:

Step1: Calculate the base area

The base is a square with side length 6 ft, so the area of the base is $6\times6 = 36$ square feet.

Step2: Calculate the lateral surface area

The total surface area is 144 square feet, so the lateral surface area (area of the four triangular faces) is $144 - 36 = 108$ square feet.

Step3: Calculate the area of one triangular face

Since there are four identical triangular faces, the area of one triangular face is $\frac{108}{4} = 27$ square feet.

Step4: Relate the area of the triangular face to its base and height

The base of each triangular face is 6 ft (same as the side of the square base), and the height of each triangular face is $x$ ft. The area of a triangle is $\frac{1}{2}\times base\times height$, so we have $\frac{1}{2}\times6\times x = 27$.

Step5: Solve for $x$

Simplifying the equation: $3x = 27$, so $x = \frac{27}{3} = 9$? Wait, no, wait. Wait, maybe I made a mistake. Wait, let's check again. Wait, the base is 6x6, so base area is 36. Total surface area is 144, so lateral surface area is 144 - 36 = 108. Four triangular faces, so each has area 108/4 = 27. The formula for the area of a triangle is (1/2)baseheight. So (1/2)6x = 27. Then 3x = 27, so x = 9? But wait, the options include 9 feet. Wait, but let's check again. Wait, maybe the base of the triangle is 6, and the height is x. So (1/2)6x = 27. So 3x = 27, x = 9. So the answer is 9 feet? But wait, the options are 4, 6, 9, 10. So 9 is an option. Wait, but maybe I messed up. Wait, let's recalculate. Total surface area = base area + 4(area of one triangular face). Base area = 66 = 36. So 36 + 4(0.56*x) = 144. So 36 + 12x = 144. Then 12x = 144 - 36 = 108. Then 12x = 108. So x = 108/12 = 9. Yes, that's correct. So x is 9 feet. Wait, but the options have 9 feet. So that's the answer.

Answer:

9 feet