A toy is in the shape of a cone mounted on a hemisphere of same base radius. If the volume of the toy is 231 `cm^3` and its diameter is 7 cm, find the height of the toy

Answer

To find the height of the toy, which consists of a cone mounted on a hemisphere, we will follow these steps:

### Step 1: Determine the radius of the base
The diameter of the toy is given as 7 cm. Therefore, the radius \( r \) can be calculated as:
\[ r = \frac{\text{diameter}}{2} = \frac{7 \, \text{cm}}{2} = 3.5 \, \text{cm} \]

**Hint:** Remember that the radius is half of the diameter.

### Step 2: Write the formula for the volume of the toy
The total volume \( V \) of the toy is the sum of the volume of the cone \( V_1 \) and the volume of the hemisphere \( V_2 \):
\[ V = V_1 + V_2 \]
Where:
- Volume of the cone \( V_1 = \frac{1}{3} \pi r^2 h \)
- Volume of the hemisphere \( V_2 = \frac{2}{3} \pi r^3 \)

### Step 3: Substitute the known values into the volume equation
We know the total volume \( V = 231 \, \text{cm}^3 \) and we can substitute the expressions for \( V_1 \) and \( V_2 \):
\[ 231 = \frac{1}{3} \pi r^2 h + \frac{2}{3} \pi r^3 \]

### Step 4: Factor out common terms
Factoring out \( \frac{1}{3} \pi \) from both terms gives:
\[ 231 = \frac{1}{3} \pi \left( r^2 h + 2r^3 \right) \]

### Step 5: Solve for \( h \)
Rearranging the equation to isolate \( h \):
\[ h + 2r = \frac{231 \times 3}{\pi r^2} \]
Thus,
\[ h = \frac{231 \times 3}{\pi r^2} - 2r \]

### Step 6: Substitute the value of \( r \) and \( \pi \)
Using \( r = 3.5 \, \text{cm} \) and \( \pi \approx \frac{22}{7} \):
\[ h = \frac{231 \times 3}{\frac{22}{7} \times (3.5)^2} - 2(3.5) \]

### Step 7: Calculate \( r^2 \)
Calculating \( r^2 \):
\[ (3.5)^2 = 12.25 \]

### Step 8: Substitute \( r^2 \) into the equation
Now substituting \( r^2 \) back:
\[ h = \frac{231 \times 3 \times 7}{22 \times 12.25} - 7 \]

### Step 9: Calculate the first term
Calculating \( \frac{231 \times 3 \times 7}{22 \times 12.25} \):
\[ = \frac{231 \times 21}{22 \times 12.25} = \frac{4851}{270.5} \approx 17.93 \]

### Step 10: Final calculation for \( h \)
Now substituting back to find \( h \):
\[ h = 17.93 - 7 \approx 10.93 \, \text{cm} \]

### Step 11: Total height of the toy
The total height of the toy is the height of the cone plus the radius of the hemisphere:
\[ \text{Total height} = h + r = 10.93 + 3.5 = 14.43 \, \text{cm} \]

### Final Answer
The height of the toy is approximately \( 14.43 \, \text{cm} \).

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