Answer :
Certainly! Let's tackle each part of the problem step-by-step.
### Part A
The given formula for the volume of a cone is:
[tex]\[ V = \frac{1}{3}\pi r^2 h \][/tex]
We need to solve this formula for [tex]\( h \)[/tex]. Follow these steps:
1. Multiply both sides by 3 to clear the fraction:
[tex]\[ 3V = \pi r^2 h \][/tex]
2. Divide both sides by [tex]\(\pi r^2\)[/tex] to isolate [tex]\( h \)[/tex]:
[tex]\[ h = \frac{3V}{\pi r^2} \][/tex]
So, the formula for [tex]\( h \)[/tex] is:
[tex]\[ h = \frac{3V}{\pi r^2} \][/tex]
### Part B
Given:
- Volume, [tex]\( V = 40 \, \text{m}^3 \)[/tex]
- Radius, [tex]\( r = 4 \, \text{m} \)[/tex]
We need to find the height [tex]\( h \)[/tex] using the formula derived in Part A. Plug the given values into the formula:
[tex]\[ h = \frac{3 \cdot 40}{\pi \cdot 4^2} \][/tex]
Now, simplify the formula step-by-step:
1. Compute the denominator:
[tex]\[ \pi \cdot 4^2 = \pi \cdot 16 \][/tex]
2. Compute the numerator:
[tex]\[ 3 \cdot 40 = 120 \][/tex]
3. Combine the results:
[tex]\[ h = \frac{120}{16\pi} \][/tex]
4. Simplify the fraction:
[tex]\[ h = \frac{120}{16\pi} = \frac{15}{2\pi} \approx \frac{15}{6.2832} \approx 2.3874 \][/tex]
Finally, round to the nearest tenth:
[tex]\[ h \approx 2.4 \, \text{m} \][/tex]
### Conclusion
- Part A: The formula for [tex]\( h \)[/tex] is:
[tex]\[ h = \frac{3V}{\pi r^2} \][/tex]
- Part B: The height of the cone, to the nearest tenth, is:
[tex]\[ h \approx 2.4 \, \text{m} \][/tex]
### Part A
The given formula for the volume of a cone is:
[tex]\[ V = \frac{1}{3}\pi r^2 h \][/tex]
We need to solve this formula for [tex]\( h \)[/tex]. Follow these steps:
1. Multiply both sides by 3 to clear the fraction:
[tex]\[ 3V = \pi r^2 h \][/tex]
2. Divide both sides by [tex]\(\pi r^2\)[/tex] to isolate [tex]\( h \)[/tex]:
[tex]\[ h = \frac{3V}{\pi r^2} \][/tex]
So, the formula for [tex]\( h \)[/tex] is:
[tex]\[ h = \frac{3V}{\pi r^2} \][/tex]
### Part B
Given:
- Volume, [tex]\( V = 40 \, \text{m}^3 \)[/tex]
- Radius, [tex]\( r = 4 \, \text{m} \)[/tex]
We need to find the height [tex]\( h \)[/tex] using the formula derived in Part A. Plug the given values into the formula:
[tex]\[ h = \frac{3 \cdot 40}{\pi \cdot 4^2} \][/tex]
Now, simplify the formula step-by-step:
1. Compute the denominator:
[tex]\[ \pi \cdot 4^2 = \pi \cdot 16 \][/tex]
2. Compute the numerator:
[tex]\[ 3 \cdot 40 = 120 \][/tex]
3. Combine the results:
[tex]\[ h = \frac{120}{16\pi} \][/tex]
4. Simplify the fraction:
[tex]\[ h = \frac{120}{16\pi} = \frac{15}{2\pi} \approx \frac{15}{6.2832} \approx 2.3874 \][/tex]
Finally, round to the nearest tenth:
[tex]\[ h \approx 2.4 \, \text{m} \][/tex]
### Conclusion
- Part A: The formula for [tex]\( h \)[/tex] is:
[tex]\[ h = \frac{3V}{\pi r^2} \][/tex]
- Part B: The height of the cone, to the nearest tenth, is:
[tex]\[ h \approx 2.4 \, \text{m} \][/tex]
