Let's start with the formula for the volume of a cone:
[tex]\[ V = \frac{1}{3} \pi r^2 h \][/tex]
Given:
- Volume [tex]\( V = 88 \pi \)[/tex] cubic feet
- Height [tex]\( h = 8 \)[/tex] feet
Step 1: Begin with the cone volume formula and substitute the given values.
[tex]\[ 88 \pi = \frac{1}{3} \pi r^2 \times 8 \][/tex]
Step 2: Simplify the equation by eliminating [tex]\(\pi\)[/tex] from both sides:
[tex]\[ 88 = \frac{8}{3} r^2 \][/tex]
Step 3: Solve for [tex]\( r^2 \)[/tex]:
[tex]\[ 88 = \frac{8}{3} r^2 \][/tex]
Multiply both sides by [tex]\(\frac{3}{8}\)[/tex]:
[tex]\[ r^2 = \frac{3 \times 88}{8} \][/tex]
[tex]\[ r^2 = \frac{264}{8} \][/tex]
[tex]\[ r^2 = 33 \][/tex]
Step 4: Calculate the value of [tex]\( r \)[/tex] by taking the square root of 33:
[tex]\[ r = \sqrt{33} \][/tex]
[tex]\[ r \approx 5.74 \][/tex]
Now let's analyze Fatima's error:
- Fatima correctly started with the volume formula and substituted the given values.
- In step 3, she should have computed [tex]\(\frac{8}{3} \times 88\)[/tex] as follows:
[tex]\[ \frac{8}{3} \times 88 = \frac{704}{3} \approx 234.67 \][/tex]
However, Fatima made an arithmetic error when simplifying this calculation, leading her to:
[tex]\[ r^2 = 234.7 \][/tex]
Consequently, she calculated:
[tex]\[ r = \sqrt{234.7} \][/tex]
[tex]\[ r \approx 15.3 \][/tex]
In summary, Fatima correctly set up the equation but made an arithmetic error in step 4, wherein she incorrectly evaluated [tex]\(\frac{8}{3} \times 88\)[/tex]. The correct value for [tex]\( r^2 \)[/tex] is 33, not 234.7. Therefore, the correct radius [tex]\( r \approx 5.74 \)[/tex] feet, not [tex]\( 15.3 \)[/tex] feet.
