To determine how long it will take for the height of the cone-shaped sand pile to increase by 1 inch, we need to go through the following steps:
1. Understand the Given Data:
- Initial height of the cone, [tex]\( h_{initial} = 20 \)[/tex] feet.
- Diameter of the base of the cone, [tex]\( d = 50 \)[/tex] feet. Therefore, the radius [tex]\( r = \frac{d}{2} = 25 \)[/tex] feet.
- Volume flow rate of sand from the funnel, [tex]\( Q = 2.5 \)[/tex] cubic feet per second.
- Increase in height, [tex]\( \Delta h = \frac{1}{12} \)[/tex] feet (since 1 inch is [tex]\(\frac{1}{12}\)[/tex] of a foot).
2. Calculate the Initial Volume of the Cone:
To find the initial volume of the cone-shaped pile, we use the formula for the volume [tex]\( V \)[/tex] of a cone:
[tex]\[
V_{initial} = \frac{1}{3} \pi r^2 h_{initial}
\][/tex]
Substituting the values:
[tex]\[
V_{initial} = \frac{1}{3} \pi (25^2) (20)
\][/tex]
Calculating it gives:
[tex]\[
V_{initial} \approx 13089.97 \text{ cubic feet}
\][/tex]
3. Calculate the Final Volume of the Cone:
The final height of the cone after the increase by 1 inch is:
[tex]\[
h_{final} = h_{initial} + \Delta h = 20 + \frac{1}{12}
\][/tex]
Now, calculate the final volume [tex]\( V_{final} \)[/tex]:
[tex]\[
V_{final} = \frac{1}{3} \pi r^2 h_{final}
\][/tex]
By substituting the values, we get:
[tex]\[
V_{final} = \frac{1}{3} \pi (25^2) \left( 20 + \frac{1}{12} \right)
\][/tex]
Calculating it gives:
[tex]\[
V_{final} \approx 13144.51 \text{ cubic feet}
\][/tex]
4. Determine the Volume of Sand Added:
The volume of sand added to increase the height by 1 inch is the difference between the final volume and the initial volume:
[tex]\[
V_{added} = V_{final} - V_{initial}
\][/tex]
[tex]\[
V_{added} \approx 13144.51 - 13089.97 = 54.54 \text{ cubic feet}
\][/tex]
5. Calculate the Time Required:
To find the time [tex]\( T \)[/tex] required to add this volume of sand at the given rate, we use the formula:
[tex]\[
T = \frac{V_{added}}{Q}
\][/tex]
Substituting the values:
[tex]\[
T = \frac{54.54}{2.5}
\][/tex]
Thus, the time required is:
[tex]\[
T \approx 21.82 \text{ seconds}
\][/tex]
Hence, it will take approximately 21.82 seconds for the height of the sand pile to increase by 1 inch.
