Sand is poured onto a level piece of ground at the rate of 0.25 m^3/min and forms a conical pile whose height is equal to its base diameter. How fast is the height increasing at the instant when the height is 0.50 m? V = 1/3(pi)(r^2)h
I got 3/pie, is that correct?
I got 4/π
let the radius be r, then the height is 2r
V= (1/3)πr^2 h
= (1/3)πr^2(2r)
= (2/3)πr^3
dV/dt = 2π r^2 dr/dt
when 2r=.5
r = .25 and dV/dt = .25
1/4 = 2π(1/16) dr/dt
times 8
2 = π dr/dt
dr/dt = 2/π
d(2r)/dt = d(height)/dt = 4/π
Thank you!
Where did you get 8 from?
To find how fast the height is increasing at a given instant, we need to determine the rate of change of height with respect to time. Let's denote the height of the cone as h (in meters) and the radius of the base as r (in meters).
From the problem statement, we know that the height of the cone is equal to its base diameter, which means h = 2r.
We are given that sand is poured onto the ground at a rate of 0.25 m^3/min. This represents the rate of change of volume with respect to time, dV/dt.
To relate the volume, V, to the radius, r, and height, h, we can use the formula for the volume of a cone:
V = (1/3)πr^2h
We want to find dh/dt, the rate of change of height with respect to time. To do that, we need to differentiate both sides of the equation with respect to time. But before that, let's express V in terms of r and h to make it easier to differentiate.
Using h = 2r, we can substitute this value into the equation for V:
V = (1/3)πr^2(2r)
V = (2/3)πr^3
Now, differentiate both sides of the equation with respect to time:
dV/dt = (2/3)π(3r^2)(dr/dt)
Since dV/dt is given as 0.25 m^3/min, and we want to find dh/dt, we need to express dV/dt in terms of dh/dt:
0.25 = (2/3)π(3r^2)(dr/dt)
We know from the problem statement that h = 0.5 m. Since h = 2r, substituting this value into the equation gives:
0.25 = (2/3)π(3(0.5^2))(dr/dt)
Simplifying further:
0.25 = (2/3)π(3(0.25))(dr/dt)
0.25 = (2/3)π(0.75)(dr/dt)
Now solve for dr/dt:
dr/dt = (0.25) / [(2/3)π(0.75)]
dr/dt = 0.25 / [(1.5/3)π]
dr/dt = 0.25 / [(1/2)π]
dr/dt = 0.25 / (π/2)
dr/dt = 0.5 / π
So, the rate at which the height is increasing at the instant when the height is 0.50 m is 0.5 / π or approximately 0.159 m/min.
Therefore, your answer of 3/π is incorrect. The correct value is approximately 0.159.