To find the total volume of the ice cream cone topped by a hemisphere, we will calculate the volume of each part separately and then sum them.
Volume of the Cone
The volume \( V \) of a right circular cone is given by the formula:
\[ V_{cone} = \frac{1}{3} \pi r^2 h \]
Where:
- \( r \) is the radius
- \( h \) is the height
Given:
- \( r = 7 \) cm
- \( h = 8 \) cm
Substituting the values:
\[ V_{cone} = \frac{1}{3} \times 3.14 \times (7)^2 \times (8) \]
Calculating:
- Calculate \( (7)^2 = 49 \)
- Then:
\[ V_{cone} = \frac{1}{3} \times 3.14 \times 49 \times 8 \]
-
Calculate \( 49 \times 8 = 392 \)
-
So now:
\[ V_{cone} = \frac{1}{3} \times 3.14 \times 392 \]
-
Calculate \( \frac{1}{3} \times 392 \approx 130.67 \)
-
Finally, calculate:
\[ V_{cone} \approx 130.67 \times 3.14 \approx 409.12 , \text{cm}^3 \]
Volume of the Hemisphere
The volume \( V \) of a hemisphere is given by the formula:
\[ V_{hemisphere} = \frac{2}{3} \pi r^3 \]
Using the same radius \( r = 7 \) cm, we find:
\[ V_{hemisphere} = \frac{2}{3} \times 3.14 \times (7)^3 \]
Calculating \( (7)^3 = 343 \):
\[ V_{hemisphere} = \frac{2}{3} \times 3.14 \times 343 \]
Calculating step by step:
-
Compute \( \frac{2}{3} \times 343 \approx 228.67 \)
-
Now calculate:
\[ V_{hemisphere} \approx 228.67 \times 3.14 \approx 718.8 , \text{cm}^3 \]
Total Volume
Now, we sum the volumes of the cone and the hemisphere:
\[ V_{total} = V_{cone} + V_{hemisphere} \approx 409.12 + 718.8 = 1127.92 , \text{cm}^3 \]
Rounding
Rounding \( V_{total} \) to the nearest tenth gives:
\[ \text{Total Volume} \approx 1127.9 , \text{cm}^3 \]
Thus, the total volume of the ice cream cone topped by a hemisphere of ice cream is approximately 1127.9 cm³.
1 answer