Question

what is the total surface area? select one: a. 2384 cm² b. 2270 cm² c. 2284 cm² d. 2168 cm²

Explanation:

Step1: Calculate surface - area of rectangular prism

The surface - area formula of a rectangular prism is $2(lw + lh+wh)$. Here, $l = 30$ cm, $w = 20$ cm, $h = 8$ cm. So, $S_{prism}=2(30\times20 + 30\times8+20\times8)=2(600 + 240 + 160)=2\times1000 = 2000$ $cm^{2}$. But we need to subtract the area of the part where the triangular prism is placed on it. The area of the contact surface is $30\times12=360$ $cm^{2}$. So the net surface - area of the rectangular prism part is $S_{1}=2000-360 = 1640$ $cm^{2}$.

Step2: Calculate surface - area of triangular prism

The two triangular faces: Area of a triangle is $A=\frac{1}{2}bh$. Here, $b = 12$ cm, $h = 8$ cm. So the combined area of two triangular faces is $2\times\frac{1}{2}\times12\times8=96$ $cm^{2}$.
The three rectangular faces of the triangular prism: One face has dimensions $30\times9 = 270$ $cm^{2}$, another has $30\times10 = 300$ $cm^{2}$, and the third (the contact - face which we already subtracted from the rectangular prism) has $30\times12$ $cm^{2}$. So the area of the non - contact rectangular faces of the triangular prism is $270+300 = 570$ $cm^{2}$.
The total surface - area of the triangular prism part is $S_{2}=96 + 570=666$ $cm^{2}$.

Step3: Calculate total surface area

The total surface area $S=S_{1}+S_{2}=1640 + 644=2284$ $cm^{2}$.

Answer:

C. $2284$ $cm^{2}$