To find the surface area of the prism with an isosceles trapezoid base, we will first calculate the area of the trapezoidal base and then the lateral surface area.
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Calculate the area of the trapezoid: The area \( A \) of a trapezoid is given by 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. Here, \( b_1 = 4 , \text{in} \), \( b_2 = 10 , \text{in} \), and \( h = 4 , \text{in} \).
Plugging in the values: \[ A = \frac{1}{2} \times (4 + 10) \times 4 = \frac{1}{2} \times 14 \times 4 = 28 , \text{in}^2 \]
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Calculate the lateral surface area of the prism: The lateral surface area of a prism is given by the perimeter of the base multiplied by the height of the prism.
First, we need to calculate the perimeter \( P \) of the trapezoid:
- The lengths of the two bases are 4 in. and 10 in.
- The legs of the trapezoid can be calculated. Since it's an isosceles trapezoid, the legs are equal. The height of the trapezoid is 4 in. We can find the lengths of the legs using the Pythagorean theorem.
The distance between the two bases is \( \frac{10 - 4}{2} = 3 , \text{in} \). Each leg \( l \) can be found using: \[ l = \sqrt{(3)^2 + (4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5 , \text{in} \]
Now, we can find the perimeter \( P \): \[ P = b_1 + b_2 + 2l = 4 + 10 + 2 \times 5 = 4 + 10 + 10 = 24 , \text{in} \]
Now, we can calculate the lateral surface area: \[ \text{Lateral Surface Area} = P \times \text{height of the prism} = 24 \times 10 = 240 , \text{in}^2 \]
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Calculate the total surface area of the prism: The total surface area \( S \) consists of the lateral surface area plus the area of the two bases: \[ S = \text{Lateral Surface Area} + 2 \times \text{Area of one base} \] Substituting the values: \[ S = 240 + 2 \times 28 = 240 + 56 = 296 , \text{in}^2 \]
Therefore, the surface area of the prism is \( \text{296 in}^2 \).
The correct answer is 296 in. squared.