To calculate the surface area of a prism with an isosceles trapezoid base, we can use the following formula for the surface area, which consists of the area of the two bases and the area of the lateral faces.
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Calculate the 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 of the trapezoid.
For this trapezoid:
- \( b_1 = 4 \) in
- \( b_2 = 10 \) in
- \( h = 4 \) in
Plugging these values into the formula: \[ A = \frac{1}{2} \times (4 + 10) \times 4 = \frac{1}{2} \times 14 \times 4 = \frac{56}{2} = 28 , \text{in}^2 \]
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Calculate the area of the two trapezoidal bases.
Since there are two bases: \[ \text{Area of the two bases} = 2 \times 28 = 56 , \text{in}^2 \]
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Calculate the lateral surface area.
The lateral surface area can be calculated by finding the perimeter of the base trapezoid and multiplying it by the height of the prism.
First, we need the lengths of the non-parallel sides (the legs of the trapezoid). For an isosceles trapezoid, the legs can be found using the Pythagorean theorem. To find the leg length:
- The top base is \( b_1 = 4 , \text{in} \) and the bottom base is \( b_2 = 10 , \text{in} \).
- The difference in lengths of the bases divided by 2 gives you the horizontal distance for each leg: \[ \text{Half difference} = \frac{b_2 - b_1}{2} = \frac{10 - 4}{2} = 3 , \text{in} \]
- The full height of the trapezoid is \( 4 , \text{in} \).
- Using the Pythagorean theorem, we find the leg length \( l \): \[ l = \sqrt{(4^2) + (3^2)} = \sqrt{16 + 9} = \sqrt{25} = 5 , \text{in} \]
Now, calculate the perimeter of the trapezoid: \[ P = b_1 + b_2 + 2l = 4 + 10 + 2 \times 5 = 4 + 10 + 10 = 24 , \text{in} \]
Finally, calculate the lateral surface area: \[ \text{Lateral area} = P \times \text{height of prism} = 24 \times 10 = 240 , \text{in}^2 \]
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Calculate the total surface area of the prism.
Now, add the area of the two bases and the lateral area: \[ \text{Total Surface Area} = \text{Area of two bases} + \text{Lateral Area} = 56 + 240 = 296 , \text{in}^2 \]
Thus, the correct surface area of the prism is 296 in². The answer response you should select is:
296 in².