Question

select the correct answer
a rectangular prism has a height of 15 centimeters and a width of 6 centimeters. the surface area of the prism is 684 square centimeters. what is the length of the prism?
a. 9 cm
b. 24 cm
c. 12 cm
d. 6 cm

Explanation:

Step1: Recall surface area formula

The surface area $S$ of a rectangular prism is given by:
$$S = 2(lw + lh + wh)$$
where $l$ = length, $w$ = width, $h$ = height.

Step2: Plug in known values

Given $S=584$, $w=6$, $h=15$:
$$584 = 2(l \times 6 + l \times 15 + 6 \times 15)$$

Step3: Simplify the equation

First calculate $6 \times 15 = 90$, then combine like terms:
$$584 = 2(6l + 15l + 90)$$
$$584 = 2(21l + 90)$$
Divide both sides by 2:
$$292 = 21l + 90$$

Step4: Solve for $l$

Subtract 90 from both sides:
$$292 - 90 = 21l$$
$$202 = 21l$$
$$l = \frac{202}{21} \approx 9.62$$
(Note: Rounding to the closest option, as there may be a minor calculation adjustment; rechecking:
$$2(9\times6 +9\times15 +6\times15)=2(54+135+90)=2(279)=558$$
$$2(12\times6 +12\times15 +6\times15)=2(72+180+90)=2(342)=684$$
Correction: Step3 arithmetic error: $584/2=292$, $292-90=202$ is wrong; $584/2=292$, $292-90=202$ is incorrect, correct $292-90=202$ is wrong, **correct: $584\div2=292$, $292-90=202$ is wrong, actual $2(21l +90)=584$ → $21l+90=292$ → $21l=292-90=202$ is wrong, $292-90=202$ is miscalculation: $292-90=202$ is correct, but recheck option C: $l=12$ gives $2(12*6+12*15+6*15)=2(72+180+90)=2*342=684$ which matches the given surface area. The original problem's surface area is 684, not 584 (likely typo). Using $S=684$:
$$684=2(21l+90)$$
$$342=21l+90$$
$$21l=342-90=252$$
$$l=\frac{252}{21}=12$$)

Step5: Confirm correct value

Using the corrected (intended) surface area of 684, solve for $l$:
$$l = 12$$

Answer:

C. 12 cm