Question

find the error alex and sia are calculating the surface area of the rectangular prism shown. is either of them correct? explain your reasoning.
alex
(5\cdot3)\cdot6 faces
= 90 in$^2$
sia
2(5\cdot4\cdot3)
= 120 in$^3$
select choice
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the surface area is
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the sum of the areas of the
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side of the prism or
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, which is
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square inches.

Explanation:

Step1: Recall Surface Area Formula

The surface area \( SA \) of a rectangular prism with length \( l \), width \( w \), height \( h \) is \( SA = 2(lw + lh + wh) \). Here, \( l = 5 \), \( w = 4 \), \( h = 3 \).

Step2: Analyze Alex's Work

Alex used \( (5 \cdot 3) \cdot 6 \) faces. But a rectangular prism has 3 pairs of congruent faces, not all 6 faces with area \( 5 \cdot 3 \). So his method is wrong.

Step3: Analyze Sia's Work

Sia used \( 2(5 \cdot 4 \cdot 3) \), but the formula is \( 2(lw + lh + wh) \), not \( 2(lwh) \). Also, her unit is \( \text{in}^3 \) (volume unit), but surface area should be in \( \text{in}^2 \).

Step4: Correct Calculation

Calculate using the formula: \( SA = 2(5 \cdot 4 + 5 \cdot 3 + 4 \cdot 3) = 2(20 + 15 + 12) = 2(47) = 94 \, \text{in}^2 \). Both Alex and Sia have errors: Alex miscalculated the face areas, Sia used wrong formula and unit.

Answer:

Both Alex and Sia are incorrect. Alex's method ignores the three different face areas (used only \( 5 \cdot 3 \) for all 6 faces), and Sia used the volume formula (with \( \text{in}^3 \)) instead of surface area formula, and also miscalculated the formula structure. The correct surface area is \( 94 \, \text{square inches} \).