Two solid right cones of equal height and radii `r_(1)` and `r_(2)` are melted and made to form a solid sphere of radius R. Then the height of the cone is

Options

A. `(4R^(2))/(r_(1)^(2) + r_(2)^(2))`
B. `(4R)/(r_(1) + r_(2))`
C. `(4R^(3))/(r_(1)^(2) + r_(2)^(2))`
D. `(R^(2))/(r_(1)^(2) + r_(2)^(2))`

Answer

To solve the problem, we need to find the height of the cones after melting them to form a solid sphere. Let's break down the solution step by step.

### Step 1: Understand the volumes involved
We have two solid right cones with equal height \( h \) and radii \( r_1 \) and \( r_2 \). The volume of a cone is given by the formula:
\[ V = \frac{1}{3} \pi r^2 h \]
Thus, the volumes of the two cones can be expressed as:
\[ V_1 = \frac{1}{3} \pi r_1^2 h \]
\[ V_2 = \frac{1}{3} \pi r_2^2 h \]

### Step 2: Calculate the total volume of the cones
The total volume of the two cones is:
\[ V_{total} = V_1 + V_2 = \frac{1}{3} \pi r_1^2 h + \frac{1}{3} \pi r_2^2 h \]
Factoring out the common terms, we get:
\[ V_{total} = \frac{1}{3} \pi h (r_1^2 + r_2^2) \]

### Step 3: Volume of the sphere
The volume of the solid sphere formed by melting the cones is given by:
\[ V_{sphere} = \frac{4}{3} \pi R^3 \]

### Step 4: Set the volumes equal
Since the total volume of the cones is equal to the volume of the sphere, we can set the equations equal to each other:
\[ \frac{1}{3} \pi h (r_1^2 + r_2^2) = \frac{4}{3} \pi R^3 \]

### Step 5: Simplify the equation
We can cancel \( \frac{1}{3} \pi \) from both sides of the equation:
\[ h (r_1^2 + r_2^2) = 4 R^3 \]

### Step 6: Solve for height \( h \)
Now, we can solve for \( h \):
\[ h = \frac{4 R^3}{r_1^2 + r_2^2} \]

### Final Result
Thus, the height of the cone is:
\[ h = \frac{4 R^3}{r_1^2 + r_2^2} \]