There are two cylindrical containers of equal capacity and equal dimensions. If the radius of one of the containers is increased by 12 ft and the height of another container is increased by 12 ft, the capacity of both the container is equally increased by K cubic ft. If the actual heights of each of the containers be 4 ft, find the increased volume of each of the container.

Options

A. (A) `1680 pi cu ft`
B. (B) `2304 pi cu ft`
C. (C) `1480 pi cu ft`
D. (D) can't be determined

Answer

To solve the problem, we need to find the increased volume of each of the cylindrical containers after modifying their dimensions. Let's break down the solution step by step.

### Step 1: Define the initial volume of the cylindrical containers
Let the initial radius of each cylindrical container be \( r \) feet and the height be \( h = 4 \) feet. The formula for the volume \( V \) of a cylinder is given by:

\[ V = \pi r^2 h \]

Substituting the height:

\[ V = \pi r^2 \cdot 4 = 4\pi r^2 \]

### Step 2: Calculate the new volume of the first container after increasing the radius
For the first container, the radius is increased by 12 feet. Therefore, the new radius becomes \( r + 12 \) feet. The new volume \( V_1 \) is:

\[ V_1 = \pi (r + 12)^2 \cdot 4 \]

### Step 3: Calculate the new volume of the second container after increasing the height
For the second container, the height is increased by 12 feet. Therefore, the new height becomes \( 4 + 12 = 16 \) feet. The new volume \( V_2 \) is:

\[ V_2 = \pi r^2 \cdot 16 \]

### Step 4: Set the increased volumes equal to each other
According to the problem, the increase in volume for both containers is equal to \( K \) cubic feet. Therefore, we can write:

\[ V_1 = V + K \]
\[ V_2 = V + K \]

Since both containers have equal increased volumes, we can equate \( V_1 \) and \( V_2 \):

\[ \pi (r + 12)^2 \cdot 4 = \pi r^2 \cdot 16 \]

### Step 5: Simplify the equation
We can cancel \( \pi \) from both sides:

\[ (r + 12)^2 \cdot 4 = r^2 \cdot 16 \]

Dividing both sides by 4 gives:

\[ (r + 12)^2 = 4r^2 \]

### Step 6: Expand and solve for \( r \)
Expanding the left side:

\[ r^2 + 24r + 144 = 4r^2 \]

Rearranging the equation:

\[ 0 = 4r^2 - r^2 - 24r - 144 \]
\[ 0 = 3r^2 - 24r - 144 \]

### Step 7: Factor or use the quadratic formula
We can simplify this equation by dividing everything by 3:

\[ 0 = r^2 - 8r - 48 \]

Now, we can factor this quadratic equation:

\[ 0 = (r - 12)(r + 4) \]

Thus, the solutions for \( r \) are:

\[ r = 12 \quad \text{or} \quad r = -4 \]

Since radius cannot be negative, we have \( r = 12 \) feet.

### Step 8: Calculate the increased volume
Now, substituting \( r = 12 \) back into the volume formula for the increased volume of each container:

For the first container:

\[ V_1 = \pi (12 + 12)^2 \cdot 4 = \pi (24)^2 \cdot 4 = \pi \cdot 576 \cdot 4 = 2304\pi \]

For the second container:

\[ V_2 = \pi (12)^2 \cdot 16 = \pi \cdot 144 \cdot 16 = 2304\pi \]

Thus, the increased volume of each container is:

\[ K = V_1 - V = 2304\pi - 4\pi(12^2) = 2304\pi - 576\pi = 1728\pi \]

### Final Answer
The increased volume of each of the containers is \( 1728\pi \) cubic feet.