Question

what is the total surface area of the solid? 842 square centimeters 702 square centimeters 558 square centimeters 982 square centimeters

Explanation:

Step1: Calculate the surface area of the rectangular prism part (excluding the top face which is covered by the pyramid)

The formula for the surface area of a rectangular prism is \(2(lw + lh+wh)\). Here, \(l = 14\) cm, \(w = 10\) cm, \(h = 6\) cm. But we exclude the top - face (\(lw\)).

$$S_{prism}=2(14\times6 + 10\times6)+14\times10$$

$$=2(84 + 60)+140$$

$$=2\times144+140$$

$$=288+140=428$$

Step2: Calculate the surface area of the triangular faces of the pyramid

The pyramid has two pairs of congruent triangular faces.
For the triangular face with base \(10\) cm and slant height \(12\) cm: Area of one such triangle \(A_1=\frac{1}{2}\times10\times12 = 60\) \(cm^2\), and there are two such triangles, so \(A_{1 - total}=2\times60 = 120\) \(cm^2\)
For the triangular face with base \(14\) cm and slant height \(11\) cm: Area of one such triangle \(A_2=\frac{1}{2}\times14\times11=77\) \(cm^2\), and there are two such triangles, so \(A_{2 - total}=2\times77 = 154\) \(cm^2\)

Step3: Calculate the total surface area

The total surface area \(S=S_{prism}+A_{1 - total}+A_{2 - total}\)

$$S = 428+120 + 154$$

$$S=702$$

Answer:

702 square centimeters