Let's solve the problem step by step.
Step 1: Understanding the Problem
Given:
- The total surface area (TSA) of the cylinder is [tex]\(924 \, \text{cm}^2\)[/tex].
- The curved surface area (CSA) of the cylinder is [tex]\(\frac{2}{3}\)[/tex] of the total surface area.
We need to find the volume of the cylinder.
Step 2: Formulas
For a cylinder:
- Total Surface Area (TSA) = [tex]\(2\pi r(h + r)\)[/tex]
- Curved Surface Area (CSA) = [tex]\(2\pi rh\)[/tex]
- Volume (V) = [tex]\(\pi r^2 h\)[/tex]
Step 3: Given Relationships
We know that:
[tex]\[ \text{CSA} = \frac{2}{3} \times \text{TSA} \][/tex]
Substituting the given total surface area:
[tex]\[ \text{CSA} = \frac{2}{3} \times 924 \, \text{cm}^2 \][/tex]
[tex]\[ \text{CSA} = 616 \, \text{cm}^2 \][/tex]
Step 4: Set Up Equations
Using the CSA formula:
[tex]\[ 2\pi rh = 616 \][/tex]
Using the TSA formula:
[tex]\[ 2\pi r(h + r) = 924 \][/tex]
Step 5: Solve for Dimensions
Let's first isolate [tex]\(h\)[/tex] from the CSA equation:
[tex]\[ 2\pi rh = 616 \][/tex]
[tex]\[ h = \frac{616}{2\pi r} \][/tex]
[tex]\[ h = \frac{308}{\pi r} \][/tex]
Substituting [tex]\(h\)[/tex] into the TSA formula:
[tex]\[ 2\pi r \left( \frac{308}{\pi r} + r \right) = 924 \][/tex]
[tex]\[ 2\pi r \left( \frac{308}{\pi r} + r \right) = 924 \][/tex]
[tex]\[ 2\pi r \left( \frac{308}{\pi r} + r \right) = 924 \][/tex]
[tex]\[ 2\pi r \left( \frac{308 + \pi r^2}{\pi r} \right) = 924 \][/tex]
[tex]\[ 2 \left( 308 + \pi r^2 \right) = 924 \][/tex]
[tex]\[ 616 + 2\pi r^2 = 924 \][/tex]
[tex]\[ 2\pi r^2 = 924 - 616 \][/tex]
[tex]\[ 2\pi r^2 = 308 \][/tex]
[tex]\[ r^2 = \frac{308}{2\pi} \][/tex]
[tex]\[ r^2 = \frac{154}{\pi} \][/tex]
[tex]\[ r = \sqrt{\frac{154}{\pi}} \][/tex]
[tex]\[ r \approx 7.001 \, \text{cm} \][/tex]
Substitute [tex]\(r\)[/tex] back to find [tex]\(h\)[/tex]:
[tex]\[ h = \frac{308}{\pi \times 7.001} \][/tex]
[tex]\[ h \approx 14.003 \, \text{cm} \][/tex]
Step 6: Calculate the Volume
[tex]\[ V = \pi r^2 h \][/tex]
[tex]\[ V = \pi \left(\frac{154}{\pi}\right) \left(\frac{308}{\pi \times 7.001}\right) \][/tex]
[tex]\[ V \approx \pi (7.001)^2 (14.003) \][/tex]
[tex]\[ V \approx 2156.434 \, \text{cm}^3 \][/tex]
Therefore, the volume of the cylinder is approximately [tex]\(2156.434 \, \text{cm}^3\)[/tex].
