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 find the surface area of the tent, which is shaped like a triangular prism, we need to calculate the areas of the two triangular ends and the three rectangular sides. ### Step 1: Calculate the area of one triangular end The area \( A \) of an equilateral triangle can be calculated using the formula: \[ A = \frac{\sqrt{3}}{4} s^2 \] where \( s \) is the side length of the triangle. In this case, the side length \( s = 4 \) feet. Calculating the area: \[ A = \frac{\sqrt{3}}{4} (4)^2 = \frac{\sqrt{3}}{4} \cdot 16 = 4\sqrt{3} \text{ square feet} \] Since there are two triangular ends, the total area for both ends is: \[ \text{Total area of triangular ends} = 2 \cdot 4\sqrt{3} = 8\sqrt{3} \text{ square feet} \] ### Step 2: Calculate the area of the rectangular sides The tent has three rectangular sides. The dimensions of these rectangles are as follows: 1. Two rectangles with dimensions \( 4 \) feet (width) by \( 9 \) feet (length). 2. One rectangle with dimensions equal to the height of the triangular end and \( 9 \) feet (length). #### Step 2.1: Calculate the height of the triangular end The height \( h \) of an equilateral triangle can be calculated using the formula: \[ h = \frac{\sqrt{3}}{2} s \] Substituting \( s = 4 \): \[ h = \frac{\sqrt{3}}{2} \cdot 4 = 2\sqrt{3} \text{ feet} \] #### Step 2.2: Calculate the areas of the rectangular sides 1. Area of the two rectangles (width = 4 feet, length = 9 feet): \[ \text{Area} = 2 \cdot (4 \cdot 9) = 2 \cdot 36 = 72 \text{ square feet} \] 2. Area of the rectangle (height = \( 2\sqrt{3} \) feet, length = 9 feet): \[ \text{Area} = 2\sqrt{3} \cdot 9 = 18\sqrt{3} \text{ square feet} \] ### Step 3: Calculate the total surface area Now, we can sum the areas of the triangular ends and the rectangular sides: \[ \text{Total Surface Area} = \text{Area of triangular ends} + \text{Area of rectangular sides} \] \[ \text{Total Surface Area} = 8\sqrt{3} + 72 + 18\sqrt{3} \] Combining like terms: \[ \text{Total Surface Area} = (8\sqrt{3} + 18\sqrt{3}) + 72 = 26\sqrt{3} + 72 \text{ square feet} \] Thus, the surface area of the tent, not including the bottom, is: \[ \boxed{26\sqrt{3} + 72} \text{ square feet} \]