Question

which expression can be used to find the surface area of the trapezoidal prism? 5·2 + 5·3 + 5·4 + 2·3 + \frac{1}{2}(2·4) 5·2 + 5·3 + 5·4 + 22·3 + \frac{1}{2}(2·4) 5·2 + 5·3 + 5·4 + 5·7 + 2·3 + \frac{1}{2}(2·4) 5·2 + 5·3 + 5·4 + 5·7 + 22·3 + \frac{1}{2}(2·4)

Explanation:

Step1: Identify the lateral - face areas

The trapezoidal prism has three rectangular lateral faces with dimensions:

• One face with dimensions \(5\times2\), area \(A_1 = 5\times2\)
• One face with dimensions \(5\times3\), area \(A_2 = 5\times3\)
• One face with dimensions \(5\times4\), area \(A_3 = 5\times4\)
• And one face with dimensions \(5\times7\), area \(A_4 = 5\times7\)

Step2: Identify the trapezoid - face areas

The two trapezoid bases have the same area. The formula for the area of a trapezoid is \(A=\frac{1}{2}(b_1 + b_2)h\). Here, for the trapezoid bases, \(b_1 = 3\), \(b_2= 7\), and \(h = 2\). The area of one trapezoid base \(A_{base}=\frac{1}{2}(3 + 7)\times2=2\times3+\frac{1}{2}(2\times4)\). Since there are two trapezoid bases, the total area of the two bases is \(2[2\times3+\frac{1}{2}(2\times4)]\)

Step3: Calculate the surface area

The surface area \(S\) of the trapezoidal prism is the sum of the areas of the lateral faces and the two trapezoid bases. So \(S=5\times2 + 5\times3+5\times4 + 5\times7+2[2\times3+\frac{1}{2}(2\times4)]\)

Answer:

\(5\times2 + 5\times3+5\times4 + 5\times7+2[2\times3+\frac{1}{2}(2\times4)]\)