Problem Analysis:
The worksheet asks us to find either the
edge length or the
surface area of cubes given certain information about their
volume or
surface area. We will solve each problem step by step using the formulas for the surface area and volume of a cube.
#### Formulas:
1.
Surface Area of a Cube:
\[
\text{Surface Area} = 6a^2
\]
where \( a \) is the edge length of the cube.
2.
Volume of a Cube:
\[
\text{Volume} = a^3
\]
where \( a \) is the edge length of the cube.
---
Problem 1:
The surface area of this cube is 54 square units. What is its edge length?
#### Solution:
1. Use the surface area formula:
\[
\text{Surface Area} = 6a^2
\]
Given:
\[
6a^2 = 54
\]
2. Solve for \( a^2 \):
\[
a^2 = \frac{54}{6} = 9
\]
3. Solve for \( a \):
\[
a = \sqrt{9} = 3
\]
#### Answer:
\[
\boxed{3}
\]
---
Problem 2:
The volume of this cube is 96 cubic units. What is its surface area?
#### Solution:
1. Use the volume formula:
\[
\text{Volume} = a^3
\]
Given:
\[
a^3 = 96
\]
2. Solve for \( a \):
\[
a = \sqrt[3]{96}
\]
Note: \( 96 \) is not a perfect cube, so we leave it in this form for now.
3. Use the surface area formula:
\[
\text{Surface Area} = 6a^2
\]
Substitute \( a = \sqrt[3]{96} \):
\[
\text{Surface Area} = 6(\sqrt[3]{96})^2
\]
4. Simplify:
\[
(\sqrt[3]{96})^2 = \sqrt[3]{96^2} = \sqrt[3]{9216}
\]
Therefore:
\[
\text{Surface Area} = 6 \cdot \sqrt[3]{9216}
\]
#### Answer:
\[
\boxed{6\sqrt[3]{9216}}
\]
---
Problem 3:
The volume of this cube is 294 cubic units. What is its surface area?
#### Solution:
1. Use the volume formula:
\[
\text{Volume} = a^3
\]
Given:
\[
a^3 = 294
\]
2. Solve for \( a \):
\[
a = \sqrt[3]{294}
\]
3. Use the surface area formula:
\[
\text{Surface Area} = 6a^2
\]
Substitute \( a = \sqrt[3]{294} \):
\[
\text{Surface Area} = 6(\sqrt[3]{294})^2
\]
4. Simplify:
\[
(\sqrt[3]{294})^2 = \sqrt[3]{294^2} = \sqrt[3]{86436}
\]
Therefore:
\[
\text{Surface Area} = 6 \cdot \sqrt[3]{86436}
\]
#### Answer:
\[
\boxed{6\sqrt[3]{86436}}
\]
---
Problem 4:
The volume of this cube is 150 cubic units. What is its edge length?
#### Solution:
1. Use the volume formula:
\[
\text{Volume} = a^3
\]
Given:
\[
a^3 = 150
\]
2. Solve for \( a \):
\[
a = \sqrt[3]{150}
\]
#### Answer:
\[
\boxed{\sqrt[3]{150}}
\]
---
Final Answers:
1. \(\boxed{3}\)
2. \(\boxed{6\sqrt[3]{9216}}\)
3. \(\boxed{6\sqrt[3]{86436}}\)
4. \(\boxed{\sqrt[3]{150}}\)
Parent Tip: Review the logic above to help your child master the concept of volume maths worksheet.
