1 6) The volume V of a right circular cone of radius r and height h is V =-ar²h. Suppose the radius of a cone increases from 8 cm to 8.15 cm while the height decreases from 12 cm to 11.9 cm. Use the total differential to approximate the change in the volume of the cone.
1 6) The volume V of a right circular cone of radius r and height h is V =-ar²h. Suppose the radius of a cone increases from 8 cm to 8.15 cm while the height decreases from 12 cm to 11.9 cm. Use the total differential to approximate the change in the volume of the cone.
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![**Problem 6: Cone Volume Calculation Using Total Differential**
The formula for the volume \( V \) of a right circular cone with radius \( r \) and height \( h \) is:
\[ V = \frac{1}{3} \pi r^2 h \]
**Problem Statement:**
Suppose the radius of a cone increases from 8 cm to 8.15 cm, while the height decreases from 12 cm to 11.9 cm. Use the total differential to approximate the change in the volume of the cone.
**Solution Approach:**
To approximate the change in volume using the total differential, consider the following steps:
1. **Total Differential Formula:**
The total differential \( dV \) of the volume \( V \) is given by:
\[ dV = \frac{\partial V}{\partial r} dr + \frac{\partial V}{\partial h} dh \]
2. **Partial Derivatives:**
- The partial derivative of \( V \) with respect to \( r \) is:
\[ \frac{\partial V}{\partial r} = \frac{2}{3} \pi r h \]
- The partial derivative of \( V \) with respect to \( h \) is:
\[ \frac{\partial V}{\partial h} = \frac{1}{3} \pi r^2 \]
3. **Calculate \( dr \) and \( dh \):**
- Change in radius, \( dr = 8.15 - 8 = 0.15 \) cm.
- Change in height, \( dh = 11.9 - 12 = -0.1 \) cm.
4. **Substitute the Values:**
Substitute \( r = 8 \) cm, \( h = 12 \) cm, \( dr = 0.15 \) cm, and \( dh = -0.1 \) cm into the total differential equation.
\[ dV = \left(\frac{2}{3} \pi \times 8 \times 12\right) \times 0.15 + \left(\frac{1}{3} \pi \times 8^2\right) \times (-0.1) \]
5. **Calculate \( dV \):**
Perform the necessary calculations to find the approximate change in volume](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Faff7206f-9910-4eba-9e78-ffe8dd60a76e%2F43d8b077-4f5a-4afc-9890-f501f23e0a92%2Frdkayno_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem 6: Cone Volume Calculation Using Total Differential**
The formula for the volume \( V \) of a right circular cone with radius \( r \) and height \( h \) is:
\[ V = \frac{1}{3} \pi r^2 h \]
**Problem Statement:**
Suppose the radius of a cone increases from 8 cm to 8.15 cm, while the height decreases from 12 cm to 11.9 cm. Use the total differential to approximate the change in the volume of the cone.
**Solution Approach:**
To approximate the change in volume using the total differential, consider the following steps:
1. **Total Differential Formula:**
The total differential \( dV \) of the volume \( V \) is given by:
\[ dV = \frac{\partial V}{\partial r} dr + \frac{\partial V}{\partial h} dh \]
2. **Partial Derivatives:**
- The partial derivative of \( V \) with respect to \( r \) is:
\[ \frac{\partial V}{\partial r} = \frac{2}{3} \pi r h \]
- The partial derivative of \( V \) with respect to \( h \) is:
\[ \frac{\partial V}{\partial h} = \frac{1}{3} \pi r^2 \]
3. **Calculate \( dr \) and \( dh \):**
- Change in radius, \( dr = 8.15 - 8 = 0.15 \) cm.
- Change in height, \( dh = 11.9 - 12 = -0.1 \) cm.
4. **Substitute the Values:**
Substitute \( r = 8 \) cm, \( h = 12 \) cm, \( dr = 0.15 \) cm, and \( dh = -0.1 \) cm into the total differential equation.
\[ dV = \left(\frac{2}{3} \pi \times 8 \times 12\right) \times 0.15 + \left(\frac{1}{3} \pi \times 8^2\right) \times (-0.1) \]
5. **Calculate \( dV \):**
Perform the necessary calculations to find the approximate change in volume
Expert Solution
Step 1
We have to find the change in the volume of cone using total differential.
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