A base of a prism is an isosceles trapezoid that has one base length of 4 in., another base length of 10 in., and a height of 4 in. The height of the prism is 10 in. Which of the following correctly calculates the surface area of the prism with an isosceles trapezoid base?

Answer 1

To calculate the surface area of a prism with an isosceles trapezoid base, we will first find the area of the trapezoidal base and then the lateral surface area of the prism.

1. Area of the trapezoidal base: The area \( A \) of a trapezoid can be calculated using the formula: \[ A = \frac{1}{2} \times (b_1 + b_2) \times h \] where \( b_1 \) and \( b_2 \) are the lengths of the two bases, and \( h \) is the height.

   For our trapezoid, we have:

   • \( b_1 = 4 \) in.
   • \( b_2 = 10 \) in.
   • \( h = 4 \) in.

   Plug these values into the area formula: \[ A = \frac{1}{2} \times (4 + 10) \times 4 = \frac{1}{2} \times 14 \times 4 = 28 \text{ in}^2. \]

2. Area of the two bases: Since the prism has two identical bases, the total area of the bases is: \[ \text{Area of two bases} = 2 \times A = 2 \times 28 = 56 \text{ in}^2. \]

3. Lateral surface area: To find the lateral surface area, we need to calculate the perimeter of the trapezoidal base and multiply it by the height of the prism.

   To find the perimeter of the trapezoid, we need to determine the lengths of the two non-parallel sides (the legs of the trapezoid).

   The legs can be calculated using the Pythagorean theorem. First, we find the length of the segment connecting the midpoints of the two bases: \[ \text{Half the difference of the bases} = \frac{b_2 - b_1}{2} = \frac{10 - 4}{2} = 3 \text{ in.} \] The height of the trapezoid is already given as 4 in. Now we can find the length of the leg using Pythagoras: \[ \text{Leg length} = \sqrt{h^2 + \left(\frac{b_2 - b_1}{2}\right)^2} = \sqrt{4^2 + 3^2} = \sqrt{16 + 9} = \sqrt{25} = 5 \text{ in.} \] Each leg of the trapezoid is 5 in. The perimeter \( P \) of the trapezoidal base is: \[ P = b_1 + b_2 + 2 \times \text{leg} = 4 + 10 + 2 \times 5 = 4 + 10 + 10 = 24 \text{ in.} \]

   The lateral surface area (LSA) of the prism is given by: \[ \text{LSA} = \text{Perimeter} \times \text{height of prism} = P \times \text{height} = 24 \times 10 = 240 \text{ in}^2. \]

4. Total surface area: The total surface area \( SA \) of the prism is the sum of the area of the bases and the lateral surface area: \[ SA = \text{Area of two bases} + \text{LSA} = 56 + 240 = 296 \text{ in}^2. \]

Thus, the surface area of the prism with an isosceles trapezoid base is \( \boxed{296} \text{ in}^2 \).