The rate of change in height is 3 cm/min
The rate of change of radius is 2 cm/min
We differentiate a quantity stated in terms of a time variable to determine how quickly it changes from one to the other.
In a similar manner, we use differential calculus to determine how quickly one variable changes in relation to another dependent variable.
The volume change rate = dv / dt = 13 m³ / min
The pile's height is always equal to three-eighths of its base diameter.
The height is = h = 4m
Using the information provided, determine the height-to-radius ratio:
h = 3 / 8 d
h = 3 / 8 (2r) (Here, the base of the diameter = 2r)
h = 3r / 4 ----- (1)
Calculate the conical pile's volume:
V = 1 / 3[tex]\pi[/tex]r²h
V = 1 / 3[tex]\pi[/tex](8h / 3)² h (using equation 1)
V = 64[tex]\pi[/tex]h³ / 27
Distinguish the above expression in terms of time:
dv / dt = d / dt (64[tex]\pi[/tex]h³ / 27)
dv / dt = (64[tex]\pi[/tex] / 27) (d / dt) (h³) ---- Use constant rule
dv / dt = (64[tex]\pi[/tex] / 27) (3h²) (dh / dt) ----- (Use chain and power rules)
dv / dt = (64[tex]\pi[/tex]h² / 9) (dh / dt)
The rate of height change in this instance = dh / dt
Replace with the known values.
13m³ / min = 64[tex]\pi[/tex] (5m)² / 9 (dh / dt)
dh / dt = 9 (13m³ /min) / 64[tex]\pi[/tex](5m)²
dh / dt = 0.033 m/min = 3.33 cm/min
dh / dt ≅ 3 cm/min
Consequently, the rate at which height changes is 3 cm/min
Radius when the height is 5 meters:
5 = 3r / 4
r = 16 / 3 m
Write the expression for the conical pile's volume:
V = [tex]\pi[/tex]r²h / 3
V = [tex]\pi[/tex]r² / 3 (3r / 4) ---- Using equation (1)
V = [tex]\pi[/tex]r³ / 4
Differentiate the expression above in terms of time:
dv / dt = d / dt ([tex]\pi[/tex]h³ / 4)
dv / dt = ([tex]\pi[/tex] / 4) d / dt (r³) ----- (Use constant rule)
dv / dt = ([tex]\pi[/tex] / 4) (3r²) (dr / dt) ---- (use power and chain rules)
dv / dt = (3[tex]\pi[/tex]r² / 4) (dr / dt)
Here, the rate of radius change is dr / dt
Replace with the known values.
13 m³ / min = (3[tex]\pi[/tex] / 4) (64 / 3)² (dr / dt)
dr / dt = (13m³/min) / (3[tex]\pi[/tex]/4)(64/3)²
dr / dt = 0.0205 m/min
dr / dt = 2.05 cm/min
dr / dt ≅ 2 cm/min
Consequently, the radius's rate of change is 2 cm/min.
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