To determine the height of the cone given its radius and volume, we can use the formula for the volume of a cone:
[tex]\[ V = \frac{1}{3} \pi r^2 h \][/tex]
Where:
- [tex]\( V \)[/tex] is the volume of the cone,
- [tex]\( r \)[/tex] is the radius of the base of the cone,
- [tex]\( h \)[/tex] is the height of the cone,
- [tex]\( \pi \)[/tex] is a mathematical constant (approximately 3.14159).
Here, we need to solve for the height [tex]\( h \)[/tex]. Given the data:
- Radius [tex]\( r = 4 \)[/tex] cm,
- Volume [tex]\( V = 134 \)[/tex] cubic centimeters.
First, we rearrange the volume formula to solve for [tex]\( h \)[/tex]:
[tex]\[ h = \frac{3V}{\pi r^2} \][/tex]
Next, we substitute the given values into this formula:
[tex]\[ h = \frac{3 \times 134}{\pi \times 4^2} \][/tex]
Now, let's calculate the numbers step-by-step:
1. Calculate [tex]\( r^2 \)[/tex]:
[tex]\[ 4^2 = 16 \][/tex]
2. Calculate the denominator:
[tex]\[ \pi \times 16 \approx 3.14159 \times 16 \approx 50.265 \][/tex]
3. Calculate the numerator:
[tex]\[ 3 \times 134 = 402 \][/tex]
4. Divide the numerator by the denominator to find [tex]\( h \)[/tex]:
[tex]\[ h = \frac{402}{50.265} \approx 7.997535890367741 \][/tex]
So, the height of the cone is approximately 7.997535 cm. Looking at the options given:
- [tex]\( h \approx 8.8 \)[/tex] cm
- [tex]\( h \approx 8 \)[/tex] cm
- [tex]\( h \approx 32 \)[/tex] cm
- [tex]\( h \approx 10.6 \)[/tex] cm
The best estimate for the cone's height, based on our calculation, is:
\[ h \approx 8 \) cm
