Question
A spherical object has a diameter of 18 cm. It has a spherical inner core with a diameter of 9 cm.
What is the volume of the outer layer?
Use 3.14 to approximate pi. Round to the nearest hundredth if necessary.
Enter your answer as a decimal in the box.
What is the volume of the outer layer?
Use 3.14 to approximate pi. Round to the nearest hundredth if necessary.
Enter your answer as a decimal in the box.
Answers
GPT-4o mini
To find the volume of the outer layer of the spherical object, we need to calculate the volumes of the outer sphere and the inner core and then subtract the volume of the inner core from the volume of the outer sphere.
1. **Calculate the radius of the outer sphere:**
\[
\text{Diameter of outer sphere} = 18 \, \text{cm}
\]
\[
\text{Radius of outer sphere} = \frac{18}{2} = 9 \, \text{cm}
\]
2. **Calculate the radius of the inner core:**
\[
\text{Diameter of inner core} = 9 \, \text{cm}
\]
\[
\text{Radius of inner core} = \frac{9}{2} = 4.5 \, \text{cm}
\]
3. **Calculate the volume of the outer sphere:**
The volume \( V \) of a sphere is given by the formula:
\[
V = \frac{4}{3} \pi r^3
\]
For the outer sphere:
\[
V_{\text{outer}} = \frac{4}{3} \times 3.14 \times (9)^3
\]
First, calculate \( 9^3 \):
\[
9^3 = 729
\]
Now substitute that into the volume formula:
\[
V_{\text{outer}} = \frac{4}{3} \times 3.14 \times 729
\]
Calculate \( \frac{4}{3} \times 3.14 = 4.18666667 \):
\[
V_{\text{outer}} \approx 4.18666667 \times 729 \approx 3059.84 \, \text{cm}^3
\]
4. **Calculate the volume of the inner core:**
For the inner core:
\[
V_{\text{inner}} = \frac{4}{3} \times 3.14 \times (4.5)^3
\]
First, calculate \( 4.5^3 \):
\[
4.5^3 = 91.125
\]
Now substitute that into the volume formula:
\[
V_{\text{inner}} = \frac{4}{3} \times 3.14 \times 91.125
\]
Again calculating \( \frac{4}{3} \times 3.14 \approx 4.18666667 \):
\[
V_{\text{inner}} \approx 4.18666667 \times 91.125 \approx 381.70 \, \text{cm}^3
\]
5. **Calculate the volume of the outer layer:**
Now, subtract the volume of the inner core from the volume of the outer sphere:
\[
V_{\text{outer layer}} = V_{\text{outer}} - V_{\text{inner}}
\]
\[
V_{\text{outer layer}} \approx 3059.84 - 381.70 \approx 2678.14 \, \text{cm}^3
\]
Thus, the volume of the outer layer is approximately \(\boxed{2678.14}\) cm³.
1. **Calculate the radius of the outer sphere:**
\[
\text{Diameter of outer sphere} = 18 \, \text{cm}
\]
\[
\text{Radius of outer sphere} = \frac{18}{2} = 9 \, \text{cm}
\]
2. **Calculate the radius of the inner core:**
\[
\text{Diameter of inner core} = 9 \, \text{cm}
\]
\[
\text{Radius of inner core} = \frac{9}{2} = 4.5 \, \text{cm}
\]
3. **Calculate the volume of the outer sphere:**
The volume \( V \) of a sphere is given by the formula:
\[
V = \frac{4}{3} \pi r^3
\]
For the outer sphere:
\[
V_{\text{outer}} = \frac{4}{3} \times 3.14 \times (9)^3
\]
First, calculate \( 9^3 \):
\[
9^3 = 729
\]
Now substitute that into the volume formula:
\[
V_{\text{outer}} = \frac{4}{3} \times 3.14 \times 729
\]
Calculate \( \frac{4}{3} \times 3.14 = 4.18666667 \):
\[
V_{\text{outer}} \approx 4.18666667 \times 729 \approx 3059.84 \, \text{cm}^3
\]
4. **Calculate the volume of the inner core:**
For the inner core:
\[
V_{\text{inner}} = \frac{4}{3} \times 3.14 \times (4.5)^3
\]
First, calculate \( 4.5^3 \):
\[
4.5^3 = 91.125
\]
Now substitute that into the volume formula:
\[
V_{\text{inner}} = \frac{4}{3} \times 3.14 \times 91.125
\]
Again calculating \( \frac{4}{3} \times 3.14 \approx 4.18666667 \):
\[
V_{\text{inner}} \approx 4.18666667 \times 91.125 \approx 381.70 \, \text{cm}^3
\]
5. **Calculate the volume of the outer layer:**
Now, subtract the volume of the inner core from the volume of the outer sphere:
\[
V_{\text{outer layer}} = V_{\text{outer}} - V_{\text{inner}}
\]
\[
V_{\text{outer layer}} \approx 3059.84 - 381.70 \approx 2678.14 \, \text{cm}^3
\]
Thus, the volume of the outer layer is approximately \(\boxed{2678.14}\) cm³.