The height of a cone is 5 m. Find the height of a cone whose volume is sixteen times its volume and radius equal to its diameter.

Answer

To solve the problem step by step, let's break down the information given and use the formulas for the volume of a cone.

### Step 1: Understand the given values
We know:
- The height of the first cone (h1) = 5 m
- The volume of the second cone (V2) is 16 times the volume of the first cone (V1).
- The radius of the second cone (r2) is equal to its diameter, which means r2 = 2 * r1.

### Step 2: Write the formula for the volume of a cone
The volume (V) of a cone is given by the formula:
\[ V = \frac{1}{3} \pi r^2 h \]

### Step 3: Calculate the volume of the first cone (V1)
Let the radius of the first cone be r1. Then, the volume of the first cone (V1) can be expressed as:
\[ V1 = \frac{1}{3} \pi r1^2 h1 \]
Substituting h1 = 5 m:
\[ V1 = \frac{1}{3} \pi r1^2 (5) \]
\[ V1 = \frac{5}{3} \pi r1^2 \]

### Step 4: Calculate the volume of the second cone (V2)
Since V2 is 16 times V1:
\[ V2 = 16 \times V1 = 16 \times \frac{5}{3} \pi r1^2 \]
\[ V2 = \frac{80}{3} \pi r1^2 \]

### Step 5: Write the volume of the second cone (using r2)
For the second cone, since the radius (r2) is equal to its diameter:
\[ r2 = 2 \times r1 \]
Thus, the volume of the second cone (V2) can also be expressed as:
\[ V2 = \frac{1}{3} \pi (r2^2) h2 \]
Substituting r2 = 2 * r1:
\[ V2 = \frac{1}{3} \pi (2r1)^2 h2 \]
\[ V2 = \frac{1}{3} \pi (4r1^2) h2 \]
\[ V2 = \frac{4}{3} \pi r1^2 h2 \]

### Step 6: Set the two expressions for V2 equal to each other
Now we can set the two expressions for V2 equal to each other:
\[ \frac{80}{3} \pi r1^2 = \frac{4}{3} \pi r1^2 h2 \]

### Step 7: Simplify the equation
We can cancel out \(\frac{1}{3} \pi r1^2\) from both sides (assuming r1 is not zero):
\[ 80 = 4 h2 \]

### Step 8: Solve for h2
Now, divide both sides by 4:
\[ h2 = \frac{80}{4} = 20 \text{ m} \]

### Final Answer
The height of the second cone (h2) is 20 m.

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