Question

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. 6 inches and 18 inchesB. 8.7 inches and 26.1 inchesC. 4.35 inches and 13.05 inchesD. 3 inches and 9 inches

Answer 1

Step-by-step explanation:

We are given the following details for a triangular prism;

`\begin{gathered} \text{Surface area}=174 \\ \text{Width}=w \\ \text{Length}=3w \\ \text{Height}=5 \end{gathered}`

Note that for the rectangular base, one side is 3 times the other. Hence, if the width is w, then the length would be 3 times w.

The surface area of a rectangular prism is;

`S=2(wl+hl+hw)`

With the side lengths given we can now substitute and we'll have;

`\begin{gathered} 174=2(\lbrack w*3w\rbrack+\lbrack5*3w\rbrack+\lbrack5w\rbrack) \\ 174=2(3w^2+15w+5w) \end{gathered}`

Divide both sides by 2;

`\begin{gathered} 87=3w^2+15w+5w \\ 87=3w^2+20w \end{gathered}`

We shall now move all terms to one side of the equation;

`3w^2+20w-87=0`

We can now solve this quadratic equationwith the quadratic equation formula;

`x=\frac{-b\pm\sqrt[]{b^2-4ac}}{2a}`

Where the variables are;

`a=3,b=20,c=-87`
`w=\frac{-20\pm\sqrt[]{20^2-4(3)(-87)}}{2(3)}`
`w=\frac{-20\pm\sqrt[]{400+1044}}{6}`
`w=\frac{-20\pm\sqrt[]{1444}}{6}`
`w=(-20\pm38)/(6)`
`w=(38-20)/(6),w=(-38-20)/(6)`
`w=3,w=-9(2)/(3)`

We now have two solutions. We shall take the positive one since our side lengths cannot be a negative value.

Therefore having the width as 3, the length which is 3 times the width is 3 times 3 and that gives 9.

Therefore;

ANSWER:

`D\colon3\text{inches and 9 inches}`