To find the dimensions of a rectangular prism where the surface area is given as 288 square centimeters, and the relationships between length, width, and height are provided, follow these steps:
1. Define the relationships:
- Let [tex]\( h \)[/tex] be the height.
- The length [tex]\( l \)[/tex] is 3 times the height: [tex]\( l = 3h \)[/tex].
- The width [tex]\( w \)[/tex] is 2 centimeters longer than the height: [tex]\( w = h + 2 \)[/tex].
2. Write the surface area formula for a rectangular prism:
- The surface area [tex]\( S \)[/tex] is given by [tex]\( S = 2(lw + lh + wh) \)[/tex].
- Substitute the given relationships into the formula:
[tex]\[
S = 2[(3h)(h + 2) + (3h)(h) + (h + 2)(h)]
\][/tex]
3. Simplify the expression:
[tex]\[
S = 2[3h^2 + 6h + 3h^2 + h^2 + 2h]
\][/tex]
[tex]\[
S = 2[7h^2 + 8h]
\][/tex]
4. Set the surface area equal to 288 square centimeters and solve for [tex]\( h \)[/tex]:
[tex]\[
2(7h^2 + 8h) = 288
\][/tex]
[tex]\[
7h^2 + 8h = 144
\][/tex]
[tex]\[
7h^2 + 8h - 144 = 0
\][/tex]
5. Solve the quadratic equation [tex]\( 7h^2 + 8h - 144 = 0 \)[/tex] to find [tex]\( h \)[/tex]:
Factoring or using the quadratic formula [tex]\( h = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)[/tex], we find the positive root of the equation [tex]\( h = 4 \)[/tex].
6. Calculate the width and length using the height value:
- Height [tex]\( h = 4 \)[/tex] cm.
- Width [tex]\( w = h + 2 = 4 + 2 = 6 \)[/tex] cm.
- Length [tex]\( l = 3h = 3 \times 4 = 12 \)[/tex] cm.
Thus, the dimensions of the prism are:
[tex]\[
\begin{array}{l}
\text{width } = 6 \text{ cm} \\
\text{height } = 4 \text{ cm} \\
\text{length } = 12 \text{ cm}
\end{array}
\][/tex]
