To find the height of a cylinder, we start with the formula for the volume of a cylinder:
[tex]\[ V = \pi r^2 h \][/tex]
where:
- [tex]\( V \)[/tex] is the volume,
- [tex]\( r \)[/tex] is the radius,
- [tex]\( h \)[/tex] is the height, and
- [tex]\(\pi\)[/tex] is a constant approximately equal to [tex]\(\frac{22}{7}\)[/tex].
Given:
- Volume, [tex]\( V = 1 \frac{2}{9} \)[/tex] cubic inches,
- Radius, [tex]\( r = \frac{1}{3} \)[/tex] inches,
- [tex]\(\pi \approx \frac{22}{7}\)[/tex].
First, convert the volume from a mixed number to an improper fraction:
[tex]\[ 1 \frac{2}{9} = 1 + \frac{2}{9} = \frac{9}{9} + \frac{2}{9} = \frac{11}{9} \, \text{cubic inches} \][/tex]
Using the volume formula, we need to solve for [tex]\( h \)[/tex]:
[tex]\[ V = \pi r^2 h \][/tex]
[tex]\[ h = \frac{V}{\pi r^2} \][/tex]
Substitute the given values into the equation:
[tex]\[ h = \frac{\frac{11}{9}}{\frac{22}{7} \left(\frac{1}{3}\right)^2} \][/tex]
First, calculate the radius squared:
[tex]\[ \left(\frac{1}{3}\right)^2 = \frac{1}{9} \][/tex]
Now, substitute this into the equation:
[tex]\[ h = \frac{\frac{11}{9}}{\frac{22}{7} \cdot \frac{1}{9}} \][/tex]
Simplify the denominator:
[tex]\[ \frac{22}{7} \cdot \frac{1}{9} = \frac{22}{63} \][/tex]
Now, the equation is:
[tex]\[ h = \frac{\frac{11}{9}}{\frac{22}{63}} \][/tex]
To divide by a fraction, multiply by its reciprocal:
[tex]\[ h = \frac{11}{9} \cdot \frac{63}{22} \][/tex]
Simplify the multiplication:
[tex]\[ h = \frac{11 \cdot 63}{9 \cdot 22} \][/tex]
[tex]\[ h = \frac{693}{198} \][/tex]
Simplify the fraction:
[tex]\[ \frac{693}{198} = \frac{7}{2}\][/tex]
So, the height of the cylinder is:
[tex]\[ h = \frac{7}{2} \][/tex]
Thus, the correct height of the cylinder is:
[tex]\(\boxed{\frac{7}{2} \, \text{inches}}\)[/tex]
