The height of a right circular cylinder is three times the radius of the base. If the height were four times the radius, the volume would be 1078 cubic centimetre more than it was previously. Find the radius of the base.

Options

A. a.6 cm
B. b.5 cm
C. c.7.5 cm
D. d.7 cm

Answer

To solve the problem step by step, we will use the formula for the volume of a cylinder and the information given in the question.

### Step 1: Define the variables
Let the radius of the base of the cylinder be \( r \) cm. According to the problem, the height \( h \) of the cylinder is three times the radius. Therefore, we can express the height as:
\[ h = 3r \]

### Step 2: Write the formula for the volume of the cylinder
The volume \( V \) of a cylinder is given by the formula:
\[ V = \pi r^2 h \]
Substituting the expression for height from Step 1:
\[ V = \pi r^2 (3r) = 3\pi r^3 \]

### Step 3: Calculate the volume with the new height
If the height were four times the radius, then:
\[ h = 4r \]
Now, we can calculate the new volume \( V' \):
\[ V' = \pi r^2 (4r) = 4\pi r^3 \]

### Step 4: Set up the equation based on the volume difference
According to the problem, the difference between the new volume and the original volume is 1078 cubic centimeters:
\[ V' - V = 1078 \]
Substituting the volumes we found:
\[ 4\pi r^3 - 3\pi r^3 = 1078 \]
This simplifies to:
\[ \pi r^3 = 1078 \]

### Step 5: Solve for \( r^3 \)
To isolate \( r^3 \), divide both sides by \( \pi \):
\[ r^3 = \frac{1078}{\pi} \]

Using \( \pi \approx \frac{22}{7} \):
\[ r^3 = \frac{1078 \times 7}{22} \]
Calculating this gives:
\[ r^3 = \frac{7546}{22} = 343 \]

### Step 6: Find the radius \( r \)
Now, take the cube root of both sides to find \( r \):
\[ r = \sqrt[3]{343} = 7 \text{ cm} \]

### Conclusion
Thus, the radius of the base of the cylinder is:
\[ \boxed{7 \text{ cm}} \]