To solve this problem, we need to consider the formulas for a cylinder, which are essential to find the necessary dimensions and their ratio.
### Step 1: Understand the Formulas
1. Curved Surface Area (CSA) of a cylinder:
[tex]\[ \text{CSA} = 2 \pi r h \][/tex]
where [tex]\( r \)[/tex] is the radius and [tex]\( h \)[/tex] is the height.
2. Volume of a cylinder:
[tex]\[ V = \pi r^2 h \][/tex]
Given:
- Curved surface area [tex]\( \text{CSA} = 264 \, \text{m}^2 \)[/tex]
- Volume [tex]\( V = 924 \, \text{m}^3 \)[/tex]
### Step 2: Set Up the Equations
Using the given formulas, we can set up the following equations:
1. [tex]\( 2 \pi r h = 264 \)[/tex]
2. [tex]\( \pi r^2 h = 924 \)[/tex]
### Step 3: Solve for Radius and Height
From the first equation:
[tex]\[ 2 \pi r h = 264 \][/tex]
Dividing both sides by [tex]\( 2 \pi \)[/tex]:
[tex]\[ r h = \frac{264}{2 \pi} \][/tex]
[tex]\[ r h = \frac{132}{\pi} \][/tex]
From the second equation:
[tex]\[ \pi r^2 h = 924 \][/tex]
Isolate [tex]\( h \)[/tex]:
[tex]\[ h = \frac{924}{\pi r^2} \][/tex]
### Step 4: Substitute [tex]\( h \)[/tex] in the first equation
Substitute the value of [tex]\( h \)[/tex] from the volume equation into the first equation:
[tex]\[ r \left( \frac{924}{\pi r^2} \right) = \frac{132}{\pi} \][/tex]
Simplify:
[tex]\[ \frac{924 r}{\pi r^2} = \frac{132}{\pi} \][/tex]
Cancel out [tex]\( \pi \)[/tex] and simplify:
[tex]\[ \frac{924}{r} = 132 \][/tex]
Rearrange to solve for [tex]\( r \)[/tex]:
[tex]\[ r = \frac{924}{132} = 7 \][/tex]
So, the radius [tex]\( r = 7 \)[/tex] meters.
### Step 5: Find the Height
Substitute [tex]\( r \)[/tex] back into the equation for [tex]\( h \)[/tex]:
[tex]\[ h = \frac{924}{\pi r^2} \][/tex]
[tex]\[ h = \frac{924}{\pi (7)^2} \][/tex]
[tex]\[ h = \frac{924}{49 \pi} \][/tex]
[tex]\[ h = \frac{924}{49 \pi} = \frac{132}{\pi} \][/tex]
So, the height [tex]\( h = \frac{132}{\pi} \)[/tex] meters.
### Step 6: Find the Diameter
The diameter of the cylinder [tex]\( D = 2r \)[/tex]:
[tex]\[ D = 2 \times 7 = 14 \][/tex] meters.
### Step 7: Find the Ratio of Diameter to Height
Finally, we need to find the ratio of the diameter to the height:
[tex]\[ \text{Ratio} = \frac{D}{h} = \frac{14}{\frac{132}{\pi}} = \frac{14 \pi}{132} = \frac{14 \pi}{132} \][/tex]
Simplify the fraction:
[tex]\[ \frac{14 \pi}{132} = \frac{7 \pi}{66} \][/tex]
Therefore, the ratio of the diameter to the height is:
[tex]\[ \frac{7 \pi}{66} = \frac{7}{6} \][/tex]
So, the ratio of the diameter to the height is [tex]\(\boxed{7:6}\)[/tex].
Hence, the correct answer is:
D. [tex]\( 7: 6 \)[/tex]
