Assume that a mango and its speed , both are spherical , now if the radius of seed is ` 2/(5)` of the thickness of the pulp . The seed lies exactly at the centre of the fruit . What percent of the total volume of the mango is its pulp ?

Options

A. a. `63""(3)/(5)%`
B. b. `97.67%`
C. c. `68""(2)/(3)%`
D. d. none of these

Answer

To solve the problem, we need to find the percentage of the total volume of the mango that is occupied by the pulp. We will follow these steps:

### Step 1: Define the thickness of the pulp
Let the thickness of the pulp be denoted as \( x \).

### Step 2: Calculate the radius of the seed
According to the problem, the radius of the seed is given as \( \frac{2}{5} \) of the thickness of the pulp. Therefore, the radius of the seed \( r_s \) can be calculated as:
\[ r_s = \frac{2}{5} x \]

### Step 3: Calculate the radius of the mango
The radius of the mango \( r_m \) is the sum of the thickness of the pulp and the radius of the seed:
\[ r_m = x + r_s = x + \frac{2}{5} x = \frac{5}{5} x + \frac{2}{5} x = \frac{7}{5} x \]

### Step 4: Calculate the volume of the seed
The volume \( V_s \) of a sphere is given by the formula:
\[ V = \frac{4}{3} \pi r^3 \]
Thus, the volume of the seed is:
\[ V_s = \frac{4}{3} \pi \left(\frac{2}{5} x\right)^3 = \frac{4}{3} \pi \cdot \frac{8}{125} x^3 = \frac{32}{375} \pi x^3 \]

### Step 5: Calculate the volume of the mango
Using the radius of the mango calculated earlier, the volume \( V_m \) of the mango is:
\[ V_m = \frac{4}{3} \pi \left(\frac{7}{5} x\right)^3 = \frac{4}{3} \pi \cdot \frac{343}{125} x^3 = \frac{1372}{375} \pi x^3 \]

### Step 6: Calculate the volume of the pulp
The volume of the pulp \( V_p \) is the volume of the mango minus the volume of the seed:
\[ V_p = V_m - V_s = \frac{1372}{375} \pi x^3 - \frac{32}{375} \pi x^3 = \frac{1340}{375} \pi x^3 \]

### Step 7: Calculate the percentage of the volume of the pulp
To find the percentage of the volume of the pulp with respect to the total volume of the mango, we use the formula:
\[ \text{Percentage of pulp} = \left( \frac{V_p}{V_m} \right) \times 100 \]
Substituting the volumes we calculated:
\[ \text{Percentage of pulp} = \left( \frac{\frac{1340}{375} \pi x^3}{\frac{1372}{375} \pi x^3} \right) \times 100 = \left( \frac{1340}{1372} \right) \times 100 \]

### Step 8: Simplify and calculate the percentage
Calculating the above expression:
\[ \frac{1340}{1372} \approx 0.9766 \]
Thus,
\[ \text{Percentage of pulp} \approx 97.66\% \]

### Conclusion
The percentage of the total volume of the mango that is its pulp is approximately **97.66%**.

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