Sure, let's break down the step-by-step solution to the given volume calculation problem.
### Step 1: Understand the given formula and values
We are given the formula for volume [tex]\( V \)[/tex] as:
[tex]\[ V = \pi r^2\left(\frac{2}{3} r + h\right) \][/tex]
We also have the following given values:
- [tex]\( r = 6 \)[/tex]
- [tex]\(\pi \approx \frac{22}{7}\)[/tex]
- [tex]\( \frac{2}{3} \)[/tex]
- [tex]\( h = 5 \times 6 = 30 \)[/tex]
### Step 2: Calculate each part of the formula
#### Part 1: Calculate [tex]\(\frac{2}{3}\)[/tex]
[tex]\[ \frac{2}{3} = 0.6666666666666666 \][/tex] (as a fraction, it remains [tex]\( \frac{2}{3} \)[/tex])
#### Part 2: Calculate [tex]\( h \)[/tex]
[tex]\[ h = 5 \times 6 = 30 \][/tex]
### Step 3: Substitute the given values into the formula
First, calculate [tex]\(\frac{2}{3} r\)[/tex]:
[tex]\[ \frac{2}{3} r = \frac{2}{3} \times 6 = 4 \][/tex]
Now, substitute [tex]\(\pi\)[/tex], [tex]\(r\)[/tex], [tex]\(\frac{2}{3} r\)[/tex], and [tex]\(h\)[/tex] into the formula:
[tex]\[ V = \frac{22}{7} \times 6^2\left(4 + 30\right) \][/tex]
### Step 4: Simplify inside the parentheses
[tex]\[ 4 + 30 = 34 \][/tex]
### Step 5: Calculate [tex]\( r^2 \)[/tex]
[tex]\[ r^2 = 6^2 = 36 \][/tex]
### Step 6: Substitute all values and calculate the volume
[tex]\[ V = \frac{22}{7} \times 36 \times 34 \][/tex]
Now, let's simplify step-by-step:
[tex]\[ V = \frac{22}{7} \times 1224 \][/tex]
We need to perform the multiplication and division:
[tex]\[ 1224 \times \frac{22}{7} = \frac{1224 \times 22}{7} = \frac{26928}{7} = 3846.8571428571427 \][/tex]
### Conclusion
The numerical values we calculated are as follows:
[tex]\[ \frac{2}{3} = 0.6666666666666666 \][/tex]
[tex]\[ h = 30 \][/tex]
[tex]\[ V = 3846.8571428571427 \][/tex]
So, the volume [tex]\( V \)[/tex] calculated using the given formula and values is:
[tex]\[ 3846.8571428571427 \][/tex]
