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The surface area of a rectangular prism is 288 square centimeters. The length of the prism is 3 times the height. The width is 2 centimeters longer than the height. What are the dimensions of the prism?

[tex]$
\begin{array}{l}
\text{width} = \square \text{ cm} \\
\text{height} = \square \text{ cm} \\
\text{length} = \square \text{ cm}
\end{array}
$[/tex]


Sagot :

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]
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