A right circular cylindrical tank has the storage capacity of 38808 ml.If the radius of the base of the cylinder is three fourth of the height. What is the diameter of the base ?

Options

A. 28 cm
B. 56 cm
C. 21 cm
D. 42 cm

Answer

To solve the problem, we need to find the diameter of the base of a right circular cylindrical tank given its volume and the relationship between its radius and height.

### Step-by-Step Solution:

1. **Understand the Volume of a Cylinder**:
The formula for the volume \( V \) of a cylinder is given by:
\[ V = \pi r^2 h \]
where \( r \) is the radius and \( h \) is the height of the cylinder.

2. **Convert Volume to Cubic Centimeters**:
The volume of the tank is given as 38808 ml. Since 1 ml = 1 cm³, we have:
\[ V = 38808 \, \text{cm}^3 \]

3. **Express Height in Terms of Radius**:
We are given that the radius \( r \) is three-fourths of the height \( h \):
\[ r = \frac{3}{4}h \]

4. **Substitute Radius in Volume Formula**:
Substitute \( r \) in the volume formula:
\[ V = \pi \left(\frac{3}{4}h\right)^2 h \]
Simplifying this gives:
\[ V = \pi \left(\frac{9}{16}h^2\right) h = \frac{9\pi}{16}h^3 \]

5. **Set the Volume Equal to 38808 cm³**:
Now, we can set the volume expression equal to the given volume:
\[ \frac{9\pi}{16}h^3 = 38808 \]

6. **Solve for Height \( h \)**:
Rearranging gives:
\[ h^3 = \frac{38808 \times 16}{9\pi} \]
Calculate \( \pi \) approximately as \( 3.14 \):
\[ h^3 = \frac{38808 \times 16}{9 \times 3.14} \]
\[ h^3 = \frac{620928}{28.26} \approx 21900 \]
Taking the cube root:
\[ h \approx \sqrt[3]{21900} \approx 28.7 \, \text{cm} \]

7. **Find the Radius \( r \)**:
Now, substituting back to find \( r \):
\[ r = \frac{3}{4}h \approx \frac{3}{4} \times 28.7 \approx 21.5 \, \text{cm} \]

8. **Calculate the Diameter**:
The diameter \( d \) is twice the radius:
\[ d = 2r \approx 2 \times 21.5 \approx 43 \, \text{cm} \]

9. **Round to the Nearest Option**:
The closest option to our calculated diameter is 42 cm.

### Final Answer:
The diameter of the base of the cylindrical tank is approximately **42 cm**.