Let's go through Arnie's work step-by-step and identify the errors:
1. Formula for Volume:
[tex]\[
V = \pi r^2 h
\][/tex]
This formula is correct for the volume of a cylinder, where [tex]\(V\)[/tex] is the volume, [tex]\(r\)[/tex] is the radius, and [tex]\(h\)[/tex] is the height (or depth in this context).
2. Plug in the Values:
[tex]\[
100 = \pi (25)^2 h
\][/tex]
Here, Arnie incorrectly plugged in the diameter of the circular area instead of the radius. The diameter is given as 25 feet, so the radius [tex]\(r\)[/tex] should be:
[tex]\[
r = \frac{25}{2} = 12.5 \text{ feet}
\][/tex]
3. Correct Step-by-Step Solution:
Using the correct radius, the volume formula should be:
[tex]\[
V = \pi r^2 h
\][/tex]
Substituting [tex]\(V = 100 \text{ cubic feet}\)[/tex] and [tex]\(r = 12.5 \text{ feet}\)[/tex]:
[tex]\[
100 = \pi (12.5)^2 h
\][/tex]
[tex]\[
100 = \pi (156.25) h
\][/tex]
4. Solve for [tex]\(h\)[/tex]:
[tex]\[
h = \frac{100}{\pi \times 156.25}
\][/tex]
Simplifying further:
[tex]\[
h = \frac{100}{490.8738521234052}
\][/tex]
[tex]\[
h \approx 0.20371832715762603 \text{ feet}
\][/tex]
So, the correct depth of the layer using 100 cubic feet of rock is approximately 0.2037 feet, which can also be converted to inches (since 1 foot = 12 inches):
[tex]\[
h \approx 0.2037 \times 12 \approx 2.44 \text{ inches}
\][/tex]
Conclusion:
Arnie's errors were:
1. Using the diameter instead of the radius in his calculation.
2. Miscalculation of the height [tex]\(h\)[/tex], both deriving the formula incorrectly and the numerical computation itself.
The correct depth of the layer of rock is approximately [tex]\( 0.2037 \)[/tex] feet, or about [tex]\( 2.44 \text{ inches} \)[/tex].
