Sure! Let’s walk through solving this problem step-by-step.
1. Calculate the diameter of the cylinder:
- We know that the ratio of the diameter to the height is [tex]\( \frac{4}{3} \)[/tex].
- Given the height of the cylinder is 26.4 cm, we can use the ratio to find the diameter.
[tex]\[
\text{Diameter} = \left( \frac{4}{3} \right) \times 26.4 \, \text{cm}
\][/tex]
- Multiplying, we get:
[tex]\[
\text{Diameter} = 35.2 \, \text{cm}
\][/tex]
2. Calculate the radius of the cylinder:
- The radius is half the diameter.
[tex]\[
\text{Radius} = \frac{\text{Diameter}}{2} = \frac{35.2}{2} \, \text{cm}
\][/tex]
- This gives us:
[tex]\[
\text{Radius} = 17.6 \, \text{cm}
\][/tex]
3. Calculate the volume of the cylinder:
- The volume [tex]\( V \)[/tex] of a cylinder is calculated using the formula:
[tex]\[
V = \pi \times \text{radius}^2 \times \text{height}
\][/tex]
- Substituting the values we have:
[tex]\[
V = \pi \times (17.6 \, \text{cm})^2 \times 26.4 \, \text{cm}
\][/tex]
- Computing this, we get:
[tex]\[
V = \pi \times 309.76 \, \text{cm}^2 \times 26.4 \, \text{cm}
\][/tex]
[tex]\[
V \approx 25690.89 \, \text{cm}^3
\][/tex]
4. Round the volume to 2 significant figures:
- The volume is approximately [tex]\( 25690.89 \, \text{cm}^3 \)[/tex].
- Rounding this to 2 significant figures gives:
[tex]\[
V \approx 2.6 \times 10^4 \, \text{cm}^3
\][/tex]
So, the volume of the cylinder, rounded to 2 significant figures, is [tex]\( 2.6 \times 10^4 \, \text{cm}^3 \)[/tex].
