sand is poured from a conveyer belt onto pile at the rate of 36pi m^3/min. The ratio of the radius to height of the circular conical pile is 2:3. How fast is the height increasing 8 min after pouring began?
radius : height = 2 : 3
r/h = 2/3
r = 2h/3
V = (1/3)π r^2 h
= (1/3)π (4h^2/9)(h)
= (4/27)π h^3
dV/dt = (4/9)π h^2 dh/dt
36π = (4/9)π h^2 dh/dt
81= h^2 dh/dt
after 8 minutes the volume was 8(36π) m^3 = 288π m^3
then (4/27)π h^3 = 288π
h^3= 1944
h = 1944^(1/3) = 12.481
so back to 81= h^2 dh/dt
dh/dt = 81/12.481^2 = appr .52 m/min
thank you I got the same answer😀 but negative 😥cause I still confused if we should take the rate as negative because the question asked how fast is the height increasing?
I think yours is the right
thanks a lot 🙏
To solve this problem, we can use related rates. Let's start by assigning variables to the quantities mentioned in the problem:
Let r be the radius of the circular conical pile (in meters).
Let h be the height of the circular conical pile (in meters).
Let V be the volume of the circular conical pile (in cubic meters).
Given:
The rate of change of volume with respect to time (dV/dt) is 36π m^3/min.
The ratio of the radius to height is 2:3, which means r = (2/3)h.
We need to find the rate at which the height is increasing, dh/dt, 8 minutes after pouring began.
To solve for dh/dt, we'll need to relate the variables and differentiate the equation to find the rate of change of height with respect to time.
First, let's express the volume of the conical pile in terms of r and h:
V = (1/3)πr^2h
Since r = (2/3)h, we can rewrite the volume equation as:
V = (1/3)π((2/3)h)^2h
V = (4/27)πh^3
Now, let's differentiate this equation with respect to time (t):
dV/dt = (4/27)π(3h^2)(dh/dt)
We know that dV/dt = 36π m^3/min, so we can substitute this value into the equation:
36π = (4/27)π(3h^2)(dh/dt)
Now, we can solve for dh/dt:
dh/dt = (36π) / ((4/27)π(3h^2))
dh/dt = 27/8h^2
We want to find dh/dt when t = 8 minutes. Plugging in t = 8, we get:
dh/dt = 27/8h^2
dh/dt = 27 / (8h^2)
To find the height at t = 8, we can use the ratio of the radius to height:
r = (2/3)h
r = (2/3)(8)
r = 16/3
Now, substitute r = 16/3 into the equation for dh/dt:
dh/dt = 27 / (8(16/3)^2)
dh/dt = 27 / (8(256/9))
dh/dt = 27 / (2048/9)
dh/dt = 9(27/2048)
dh/dt = 27/227
Therefore, the height is increasing at a rate of 27/227 meters per minute, 8 minutes after pouring began.
To find the rate at which the height is increasing, we need to find an expression that relates the rate of change of the height to the rate of change of the volume of sand in the pile.
Let's start by analyzing the given information. We know that sand is poured onto the pile at a constant rate of 36π m³/min. This rate refers to the rate of change of the volume of sand in the pile.
We are also given that the ratio of the radius to the height of the circular conical pile is 2:3. This means that for every increase in height by h, the radius increases by (2/3)h.
Let's denote the height of the pile as h(t) at time t, where t is measured in minutes. Similarly, let's denote the radius as r(t) at time t.
We can find the volume of the cone-shaped pile using the formula for the volume of a cone:
V = (1/3) * π * r^2 * h
To relate the rate of change of the volume to the height, we can take the derivative of both sides with respect to time:
dV/dt = (1/3) * π * (2rh * dh/dt + r^2 * dh/dt)
Now, let's substitute the known values into the equation: r(t) = (2/3)h(t)
dV/dt = (1/3) * π * (2(2/3)h(t) * dh/dt + (2/3)h(t)^2 * dh/dt)
We are given that dV/dt = 36π, so we can substitute that in:
36π = (1/3) * π * (2(2/3)h(t) * dh/dt + (2/3)h(t)^2 * dh/dt)
Canceling the π terms and simplifying:
36 = (2/3) * (2h(t) * dh/dt + (2/3)h(t)^2 * dh/dt)
Now, let's solve for dh/dt, which represents the rate at which the height is increasing:
36 = (4/3)h(t) * dh/dt + (4/9)h(t)^2 * dh/dt
Multiplying through by 9 to eliminate the fractions:
36 * 9 = 12h(t) * dh/dt + 4h(t)^2 * dh/dt
324 = 12h(t) * dh/dt + 4h(t)^2 * dh/dt
We are given that at t = 8 min, the height is h = 0. We need to find the rate at which the height is increasing 8 min after pouring began, so we need to evaluate the equation at t = 8 min.
324 = 12 * h(8) * dh/dt + 4 * h(8)^2 * dh/dt
Since h(8) = 0, the equation simplifies to:
324 = 0 + 0 * dh/dt
Now, we have an equation where the rate of change of height, dh/dt, is multiplied by zero, which means that dh/dt is zero.
Therefore, the height is not increasing at all 8 min after pouring began.