A cylinderical chocobar has its radius r unit and height 'h' unit. If we wish to increase the volume by same unit either by increasing its radius alone or its height alone, then how many unit we have to increase the radius or height?

Options

A. `(r^2+2r)/h`
B. `(r^2-2rh)/h`
C. `(2r^2-rh)/h^2`
D. `(pir^2)/2h`

Answer

To solve the problem of how much we need to increase the radius or height of a cylindrical chocobar to increase its volume by the same unit, we can follow these steps:

### Step-by-Step Solution:

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

2. **Initial Volume**:
The initial volume of the cylinder is:
\[ V_1 = \pi r^2 h \]

3. **Increment in Radius and Height**:
Let \( x \) be the increment in both radius and height. Therefore, the new radius and height will be:
\[ r' = r + x \quad \text{and} \quad h' = h + x \]

4. **New Volume**:
The new volume \( V_2 \) after the increments will be:
\[ V_2 = \pi (r + x)^2 (h + x) \]

5. **Setting Up the Equation**:
We want the increase in volume to be the same unit, so we set up the equation:
\[ V_2 - V_1 = 1 \]
This leads to:
\[ \pi (r + x)^2 (h + x) - \pi r^2 h = 1 \]

6. **Expanding the New Volume**:
Expanding \( V_2 \):
\[ V_2 = \pi [(r^2 + 2rx + x^2)(h + x)] \]
Expanding further:
\[ V_2 = \pi [r^2h + r^2x + 2r x h + 2rx^2 + hx^2] \]

7. **Setting the Equation**:
Now, substituting back into our equation:
\[ \pi [r^2h + r^2x + 2r x h + 2rx^2 + hx^2] - \pi r^2 h = 1 \]
This simplifies to:
\[ \pi [r^2x + 2r x h + 2rx^2 + hx^2] = 1 \]

8. **Dividing by π**:
Dividing through by \( \pi \):
\[ r^2x + 2r x h + 2rx^2 + hx^2 = \frac{1}{\pi} \]

9. **Rearranging the Equation**:
Rearranging gives us:
\[ hx^2 + (2rh + r^2)x - \frac{1}{\pi} = 0 \]

10. **Using the Quadratic Formula**:
This is a quadratic equation in the form \( ax^2 + bx + c = 0 \). We can use the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \( a = h \), \( b = 2rh + r^2 \), and \( c = -\frac{1}{\pi} \).

11. **Final Calculation**:
Plugging in the values:
\[ x = \frac{-(2rh + r^2) \pm \sqrt{(2rh + r^2)^2 - 4h(-\frac{1}{\pi})}}{2h} \]

### Final Answer:
The increment \( x \) needed in both the radius and height to achieve the same unit increase in volume can be calculated using the above formula.