A conical cap just covers two spheres placed one above the other on a table. If the radii of the sphere are 1" and 2¼" find the height of the cone

Options

A. `5.3''`
B. `7.8''`
C. `8.1''`
D. `6.5''`

Answer

To find the height of the conical cap that covers two spheres, we can follow these steps:

### Step 1: Identify the radii of the spheres
The radii of the two spheres are given as:
- Radius of the first sphere (small sphere) = 1 inch
- Radius of the second sphere (large sphere) = 2¼ inches = \( \frac{9}{4} \) inches

### Step 2: Calculate the total height of the spheres
The total height of the two spheres placed one above the other can be calculated by adding their diameters:
- Diameter of the first sphere = \( 2 \times 1 = 2 \) inches
- Diameter of the second sphere = \( 2 \times \frac{9}{4} = \frac{18}{4} = 4.5 \) inches

Total height of the spheres = Diameter of first sphere + Diameter of second sphere
\[ = 2 + 4.5 = 6.5 \text{ inches} \]

### Step 3: Set up the relationship for the cone
Let \( h \) be the height of the cone from the apex to the base of the smaller sphere. The total height of the cone will be:
\[ \text{Height of cone} = h + \text{height of the spheres} \]

### Step 4: Use similar triangles to find the height of the cone
Using the properties of similar triangles, we can set up the following relationship:
- Let \( AB \) be the height of the cone from the apex to the base of the smaller sphere.
- The height from the apex of the cone to the center of the smaller sphere is \( AB = h \).
- The height from the apex of the cone to the center of the larger sphere is \( AC = h + 1 + \frac{9}{4} \).

Using the similarity of triangles \( ABE \) and \( ACF \):
\[ \frac{AB}{BE} = \frac{AC}{CF} \]
Where:
- \( BE = 1 \) (radius of the smaller sphere)
- \( CF = \frac{9}{4} \) (radius of the larger sphere)

Substituting the values:
\[ \frac{h}{1} = \frac{h + 1 + \frac{9}{4}}{\frac{9}{4}} \]

### Step 5: Solve for \( h \)
Cross-multiplying gives:
\[ h \cdot \frac{9}{4} = h + 1 + \frac{9}{4} \]
\[ \frac{9}{4}h = h + 1 + \frac{9}{4} \]
Rearranging gives:
\[ \frac{9}{4}h - h = 1 + \frac{9}{4} \]
\[ \frac{5}{4}h = \frac{13}{4} \]
\[ h = \frac{13}{5} \text{ inches} \]

### Step 6: Calculate the total height of the cone
Now we can find the total height of the cone:
\[ \text{Height of cone} = h + \frac{11}{2} \]
Converting \( \frac{11}{2} \) to a decimal:
\[ \frac{11}{2} = 5.5 \]
So,
\[ \text{Height of cone} = \frac{13}{5} + 5.5 = \frac{13}{5} + \frac{55}{10} = \frac{26}{10} + \frac{55}{10} = \frac{81}{10} = 8.1 \text{ inches} \]

### Final Answer
The height of the cone is \( 8.1 \) inches.

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