IDNLearn.com is the perfect place to get answers, share knowledge, and learn new things. Our platform is designed to provide reliable and thorough answers to all your questions, no matter the topic.
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]
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]
Your presence in our community is highly appreciated. Keep sharing your insights and solutions. Together, we can build a rich and valuable knowledge resource for everyone. Find clear and concise answers at IDNLearn.com. Thanks for stopping by, and come back for more dependable solutions.