If the radius of a spherical balloon increases by 0.1% then its volume increases approximately by

Options

A. `0.2%`
B. `0.3%`
C. `0.4%`
D. `0.05%`

Answer

To solve the problem of how much the volume of a spherical balloon increases when its radius increases by 0.1%, we can follow these steps:

### Step 1: Understand the relationship between radius and volume
The volume \( V \) of a sphere is given by the formula:
\[ V = \frac{4}{3} \pi r^3 \]
where \( r \) is the radius of the sphere.

### Step 2: Differentiate the volume with respect to the radius
To find how the volume changes with respect to the radius, we can differentiate \( V \) with respect to \( r \):
\[ \frac{dV}{dr} = 4 \pi r^2 \]

### Step 3: Use the concept of relative change
We know that if the radius increases by a small amount \( dr \), the corresponding change in volume \( dV \) can be approximated using the formula:
\[ \frac{dV}{V} \approx 3 \frac{dr}{r} \]
This is derived from the differentiation of the volume formula.

### Step 4: Substitute the given percentage change in radius
We are given that the radius increases by 0.1%. This can be expressed as:
\[ \frac{dr}{r} = \frac{0.1}{100} = 0.001 \]

### Step 5: Calculate the approximate percentage increase in volume
Now, substituting this value into the formula for the change in volume:
\[ \frac{dV}{V} \approx 3 \times \frac{dr}{r} = 3 \times 0.001 = 0.003 \]
To express this as a percentage, we multiply by 100:
\[ \frac{dV}{V} \times 100 \approx 0.003 \times 100 = 0.3\% \]

### Conclusion
Thus, the volume of the balloon increases approximately by **0.3%** when the radius increases by 0.1%.

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