Two solid right circular cones have the same height. The radii of their bases are a and b. They are melted and recast into a cylinder of same height. The radius of the base of the cylinder is

Options

A. `(a+b)/sqrt3`
B. `(a+b)/3`
C. `sqrt((a^(2) + b^(2)))/3`
D. `sqrt((a^(2) + b^(2))/3)`

Answer

To find the radius of the base of the cylinder formed by melting two solid right circular cones, we can follow these steps:

### Step 1: Understand the volumes of the cones
The volume \( V \) of a right circular cone is given by the formula:
\[ V = \frac{1}{3} \pi r^2 h \]
where \( r \) is the radius of the base and \( h \) is the height.

### Step 2: Calculate the volumes of the two cones
Let the first cone have a radius \( a \) and height \( h \). Its volume \( V_1 \) is:
\[ V_1 = \frac{1}{3} \pi a^2 h \]
Let the second cone have a radius \( b \) and the same height \( h \). Its volume \( V_2 \) is:
\[ V_2 = \frac{1}{3} \pi b^2 h \]

### Step 3: Find the total volume of the two cones
The total volume \( V_{total} \) of the two cones when melted is:
\[ V_{total} = V_1 + V_2 = \frac{1}{3} \pi a^2 h + \frac{1}{3} \pi b^2 h \]
Factoring out the common terms:
\[ V_{total} = \frac{1}{3} \pi h (a^2 + b^2) \]

### Step 4: Set up the volume of the cylinder
The volume \( V_c \) of the cylinder formed from the melted cones is given by:
\[ V_c = \pi r^2 h \]
where \( r \) is the radius of the base of the cylinder.

### Step 5: Equate the volumes
Since the volume of the melted cones equals the volume of the cylinder, we have:
\[ \frac{1}{3} \pi h (a^2 + b^2) = \pi r^2 h \]

### Step 6: Simplify the equation
We can cancel \( \pi \) and \( h \) from both sides (assuming \( h \neq 0 \)):
\[ \frac{1}{3} (a^2 + b^2) = r^2 \]

### Step 7: Solve for \( r \)
Multiplying both sides by 3 gives:
\[ a^2 + b^2 = 3r^2 \]
Now, divide by 3:
\[ r^2 = \frac{a^2 + b^2}{3} \]
Taking the square root gives:
\[ r = \sqrt{\frac{a^2 + b^2}{3}} \]

### Final Answer
The radius of the base of the cylinder is:
\[ r = \sqrt{\frac{a^2 + b^2}{3}} \]