Respuesta :
EXPLANATION:
We are given the following details for a triangular prism;
[tex]\begin{gathered} \text{Surface area}=174 \\ \text{Width}=w \\ \text{Length}=3w \\ \text{Height}=5 \end{gathered}[/tex]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;
[tex]S=2(wl+hl+hw)[/tex]With the side lengths given we can now substitute and we'll have;
[tex]\begin{gathered} 174=2(\lbrack w\times3w\rbrack+\lbrack5\times3w\rbrack+\lbrack5w\rbrack) \\ 174=2(3w^2+15w+5w) \end{gathered}[/tex]Divide both sides by 2;
[tex]\begin{gathered} 87=3w^2+15w+5w \\ 87=3w^2+20w \end{gathered}[/tex]We shall now move all terms to one side of the equation;
[tex]3w^2+20w-87=0[/tex]We can now solve this quadratic equationwith the quadratic equation formula;
[tex]x=\frac{-b\pm\sqrt[]{b^2-4ac}}{2a}[/tex]Where the variables are;
[tex]a=3,b=20,c=-87[/tex][tex]w=\frac{-20\pm\sqrt[]{20^2-4(3)(-87)}}{2(3)}[/tex][tex]w=\frac{-20\pm\sqrt[]{400+1044}}{6}[/tex][tex]w=\frac{-20\pm\sqrt[]{1444}}{6}[/tex][tex]w=\frac{-20\pm38}{6}[/tex][tex]w=\frac{38-20}{6},w=\frac{-38-20}{6}[/tex][tex]w=3,w=-9\frac{2}{3}[/tex]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:
[tex]D\colon3\text{inches and 9 inches}[/tex]
