Respuesta :
Answer:
[tex]\frac{dh}{dt}=\frac{160}{169\pi } ft/min[/tex]
Step-by-step explanation:
This is a classic related rates problem. Gotta love calculus!
Start out with the formula for the volume of a cone, which is
[tex]V=\frac{1}{3}\pi r^2h[/tex]
and with what we know, which is [tex]\frac{dV}{dt}=40[/tex]
and the fact that the diameter = height (we will come back to that in a bit).
We need to find [tex]\frac{dh}{dt}[/tex] when h = 13
The thing we need to notice now is that there is no information given to us that involves the radius. It does, however, give us a height. We need to replace the r with something in terms of h. Let's work on that first.
We know that d = h. Because d = 2r, we can say that 2r = h, and solving for r gives us that [tex]r=\frac{h}{2}[/tex].
Now we can rewrite the formula with that replacement:
[tex]V=\frac{1}{3}\pi (\frac{h}{2})^2h[/tex]
Simplify that all the way down to
[tex]V=\frac{1}{12}\pi h^3[/tex]
The derivative of that function with respect to time is
[tex]\frac{dV}{dt}=\frac{1}{12}\pi(3h^2)\frac{dh}{dt}[/tex]
Filling in what we have gives us this:
[tex]40=\frac{1}{12}\pi (3)(13)^2\frac{dh}{dt}[/tex]
Solve that for the rate of change of the height:
[tex]\frac{dh}{dt}=\frac{160}{169\pi } \frac{ft}{min}[/tex]
or in decimal form:
[tex]\frac{dh}{dt}=.95\pi \frac{ft}{min}[/tex]
This involves relationship between rates using Calculus.
dh/dt = 0.3 ft/min
- We are given;
Volumetric rate; dv/dt = 40 ft³/min
height of pile; h = 13 ft
We are not given the diameter here but as we are dealing with a right circular cone, we will assume that the diameter is equal to the height.
Thus; diameter; d = 13 ft
radius; r = h/2 = d/2 = 13/2
radius; r= 6.5 ft
- Formula for volume of a cone is;
V = ¹/₃πr²h
We want to find how fast the height is increasing and this is dh/dt.
Thus, we will need to express r in the volume formula in terms of h;
V = ¹/₃π(h/2)²h
V = ¹/₃π(h²/4)h
V = ¹/₁₂πh³
- differentiating both sides with respect to time t gives;
dV/dt = 3(¹/₁₂πh²)dh/dt
dV/dt = ¹/₄πh²(dh/dt)
Plugging in the relevant values, we have;
40 = ¹/₄π × 13² × (dh/dt)
dh/dt = (40 × 4)/(π × 13²)
dh/dt = 0.3 ft/min
Read more at; https://brainly.com/question/15585520
