Question

select the correct answer.
the surface area of a rectangular prism is 174 square inches. the rectangular base has one side length 3 times the other. the height of the prism is 5 inches. what are the maximum lengths of the sides of the base?
a. 4.35 inches and 13.05 inches
b. 3 inches and 9 inches
c. 8.7 inches and 26.1 inches
d. 6 inches and 18 inches

Explanation:

Step1: Recall surface area formula for rectangular prism

The surface area \( S \) of a rectangular prism is given by \( S = 2(lw + lh + wh) \), where \( l \) is length, \( w \) is width, and \( h \) is height. Let one side of the base be \( x \), then the other side is \( 3x \), and height \( h = 5 \) inches. So substitute into the formula:
\( 174 = 2(x(3x) + x(5) + 3x(5)) \)

Step2: Simplify the equation

First, simplify inside the parentheses: \( x(3x)=3x^{2} \), \( x(5) = 5x \), \( 3x(5)=15x \). So the equation becomes:
\( 174 = 2(3x^{2}+5x + 15x) \)
Combine like terms: \( 5x + 15x = 20x \), so:
\( 174 = 2(3x^{2}+20x) \)
Divide both sides by 2:
\( 87 = 3x^{2}+20x \)
Rearrange into standard quadratic form:
\( 3x^{2}+20x - 87 = 0 \)

Step3: Solve the quadratic equation

Use the quadratic formula \( x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a} \) for \( ax^{2}+bx + c = 0 \). Here, \( a = 3 \), \( b = 20 \), \( c=-87 \).
First, calculate the discriminant \( \Delta=b^{2}-4ac=(20)^{2}-4(3)(-87)=400 + 1044 = 1444 \)
Then, \( x=\frac{-20\pm\sqrt{1444}}{2(3)}=\frac{-20\pm38}{6} \)
We take the positive root (since length can't be negative):
\( x=\frac{-20 + 38}{6}=\frac{18}{6}=3 \)
So one side is \( x = 3 \) inches, the other is \( 3x=9 \) inches.

Answer:

B. 3 inches and 9 inches