A tent is shaped like a triangular prism. Each end of the tent is an equilateral triangle with side length 4 feet. The tent is 9 feet long. Determine the surface area of the tent, not including the bottom.

To calculate the surface area of the tent, we start by determining the area of the two equilateral triangular ends. The formula for the area of an equilateral triangle is given by: \[ \text{Area} = \frac{\sqrt{3}}{4} s^2 \] where \( s \) is the side length. For our tent: \[ \text{Area} = \frac{\sqrt{3}}{4} \times 4^2 = \frac{\sqrt{3}}{4} \times 16 = 4\sqrt{3} \text{ square feet} \] Since there are two triangular ends, their total area is: \[ 2 \times 4\sqrt{3} = 8\sqrt{3} \text{ square feet} \] Next, we find the area of the three rectangular sides. The width of the tent (the length of each side of the triangle) is 4 feet, and the height (the length of the tent) is 9 feet. The area of each rectangular side is calculated as follows: - For the two sides that are rectangles with a height of 9 feet and a width of 4 feet: \[ \text{Area} = \text{Height} \times \text{Width} = 9 \times 4 = 36 \text{ square feet} \text{ (for each side)} \] Since there are two such sides, the total area for both is: \[ 2 \times 36 = 72 \text{ square feet} \] - The third rectangular side appears at the front and back, often called the "base" of the tent. It will also have the same width and height: \[ \text{Area} = 9 \times 4 = 36 \text{ square feet} \] Finally, we add all the areas together (the top parts are not included since we’re only looking for the lateral surface area). Hence, the total surface area of the tent is: \[ \text{Total Surface Area} = \text{Area of 2 Triangles} + \text{Area of 3 Rectangles} \] \[ = 8\sqrt{3} + 72 \] So, the surface area of the tent, not including the bottom, is: \[ 8\sqrt{3} + 72 \text{ square feet} \]