In a right circular cone, the radius of its base is 7 cm and its height 24 cm. A cross-section is made through the midpoint of the height parallel to the base. The volume of the upper portion is

Options

A. `169 cm^(3)`
B. `154 cm^(3)`
C. `1078 cm^(3)`
D. `800 cm^(3)`

Answer

To find the volume of the upper portion of the cone after making a cross-section through the midpoint of its height, we can follow these steps:

### Step 1: Understand the dimensions of the cone
- The radius of the base of the cone (R) = 7 cm
- The height of the cone (H) = 24 cm
- The midpoint of the height is at H/2 = 24 cm / 2 = 12 cm.

### Step 2: Determine the dimensions of the smaller cone
Since the cross-section is made at the midpoint of the height, we need to find the dimensions of the smaller cone that is formed above this cross-section.
- The height of the smaller cone (h) = 12 cm (since it is the upper half of the original cone).
- The radius of the smaller cone (r) can be determined using the similarity of triangles. The ratio of the radius to the height of the original cone is the same as that of the smaller cone:

\[ \frac{r}{R} = \frac{h}{H} \]

Substituting the known values:

\[ \frac{r}{7} = \frac{12}{24} \]

Simplifying gives:

\[ \frac{r}{7} = \frac{1}{2} \implies r = \frac{7}{2} = 3.5 \text{ cm} \]

### Step 3: Calculate the volume of the upper cone
The volume \( V \) of a cone is given by the formula:

\[ V = \frac{1}{3} \pi r^2 h \]

Substituting the values for the smaller cone:
- \( r = 3.5 \) cm
- \( h = 12 \) cm

\[ V = \frac{1}{3} \pi (3.5)^2 (12) \]

Calculating \( (3.5)^2 \):
\[ (3.5)^2 = 12.25 \]

Now substituting this back into the volume formula:
\[ V = \frac{1}{3} \pi (12.25)(12) \]

Calculating \( 12.25 \times 12 \):
\[ 12.25 \times 12 = 147 \]

Now substituting this value back in:
\[ V = \frac{1}{3} \pi (147) \]

Using \( \pi \approx \frac{22}{7} \):
\[ V = \frac{1}{3} \times \frac{22}{7} \times 147 \]

Calculating this:
\[ V = \frac{22 \times 147}{21} = \frac{22 \times 7}{1} = 154 \text{ cm}^3 \]

### Final Answer
The volume of the upper portion of the cone is **154 cm³**.

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