Question

a triangular pyramid and its net are shown. kain uses the calculations below to conclude that the surface area of the triangular pyramid is 238.5 square inches.
3(\frac{1}{2}×10×13)+(\frac{1}{2}×10×8.7)
3(65)+43.5
238.5
is kain correct? use the drop - down menus to explain your reasoning.
click the arrows to choose an answer from each menu.
the expression 3(\frac{1}{2}×10×13)+(\frac{1}{2}×10×8.7).
choose...
in the next step, when kain equates 3(\frac{1}{2}×10×13)+(\frac{1}{2}×10×8.7) to 3(65)+43.5, he
choose...
kain choose... the surface area correctly. the surface area of the triangular pyramid is choose... square inches.

Explanation:

Step1: Recall surface - area formula for triangular pyramid

The surface area of a triangular pyramid is the sum of the areas of its four triangular faces. The formula for the area of a triangle is $A=\frac{1}{2}bh$, where $b$ is the base and $h$ is the height.

Step2: Analyze Rain's expression

The expression $3(\frac{1}{2}\times10\times13)+(\frac{1}{2}\times10\times8.7)$ should represent the sum of the areas of the four faces. The first part $3(\frac{1}{2}\times10\times13)$ seems to account for three of the faces with base $b = 10$ and height $h = 13$, and the second part $(\frac{1}{2}\times10\times8.7)$ for the fourth face with base $b = 10$ and height $h = 8.7$. But looking at the net, the height of the base - triangle should be $12$ (not $8.7$) for the correct calculation. So the expression is incorrect.

Step3: Analyze Rain's simplification

When Rain simplifies $\frac{1}{2}\times10\times13$ to $65$ and $\frac{1}{2}\times10\times8.7$ to $43.5$, the arithmetic simplification $3(\frac{1}{2}\times10\times13)+(\frac{1}{2}\times10\times8.7)=3(65)+43.5$ is correct in terms of arithmetic operations.

Step4: Calculate the correct surface area

The correct surface area should be $3(\frac{1}{2}\times10\times13)+(\frac{1}{2}\times10\times12)$.
\[

$$\begin{align*} 3(\frac{1}{2}\times10\times13)+(\frac{1}{2}\times10\times12)&=3(65)+60\\ &=195 + 60\\ &=255 \end{align*}$$

\]

Answer:

The expression $3(\frac{1}{2}\times10\times13)+(\frac{1}{2}\times10\times8.7)$ is incorrect.
In the next step, when Rain equates $3(\frac{1}{2}\times10\times13)+(\frac{1}{2}\times10\times8.7)$ to $3(65)+43.5$, he simplifies the arithmetic correctly.
Rain does not calculate the surface area correctly. The surface area of the triangular pyramid is $255$ square inches.