Question

study the example showing how to find the surface area of a three - dimensional figure. then solve problems 1 - 5. example: nina uses clay to make an earring in the shape of a square pyramid. she plans to cover the earring with glitter. what is the area nina will cover with glitter? use a net to help you find the surface area of the pyramid. the pyramid has five faces: one square base and four identical triangles. find the sum of the areas of the faces. 18·18 + 4(1/2)(18·14)=324 + 2(18·14)=324 + 2(252)=324 + 504=828 the area nina will cover with glitter is 828 mm². 1. ayana wrote the expression below. does this expression represent the surface area of nina’s earring in the example? explain. 2(18·18)+2(1/2)(18·14)+2(1/2)(18·14)

Explanation:

Step1: Analyze the surface - area components

The square - based pyramid has 1 square base with side length 18 mm and 4 identical triangular faces with base 18 mm and height 14 mm. The correct formula for the surface area is \(A = 18\times18+4\times\frac{1}{2}\times(18\times14)\).

Step2: Simplify Ayana's expression

Simplify \(2(18\times18)+2\times\frac{1}{2}\times(18\times14)+2\times\frac{1}{2}\times(18\times14)\).
\[

$$\begin{align*} 2(18\times18)+2\times\frac{1}{2}\times(18\times14)+2\times\frac{1}{2}\times(18\times14)&=2\times18^{2}+(18\times14)+(18\times14)\\ &=2\times324 + 252+252\\ &=648+252 + 252\\ &=1152 \end{align*}$$

\]

Step3: Compare with the correct formula

The correct surface - area formula \(18\times18 + 4\times\frac{1}{2}\times(18\times14)=324 + 504=828\). Ayana's expression gives a different result. The term \(2(18\times18)\) implies two square bases instead of one, which is incorrect.

Answer:

No, Ayana's expression does not represent the surface area of Nina's earring. The expression \(2(18\times18)\) has an extra square - base term, making the calculation incorrect. The correct surface area is 828 \(mm^{2}\), while Ayana's expression evaluates to 1152.