A vertical container with base area of length L and width W is being filled with identical pieces of candy, each with a volume of v and a mass m. Assume that the volume of the empty spaces between the candies is negligible. If the height of the candies in the container increases at a certain rate per unit time dH/dt, at what rate per unit time does the mass of the candies in the container increase?
d volume/d h = base area (draw a picture :)
dv/dt = dv/dh * dh/dt
so
dv/dt = base area * dh/dt
let rho = m/v of candy
d m = dv (rho) so dv = dm/rho
so
(1/rho) dm/dt = base area * dh/dt
or
dm/dt = rho * base area * dh/dt
= (m/v) * L * W * dh/dt
Did you ever get the correct answer? Im on this right now...
Thanks for the help!
I also suspected this as the answer, but unfortunately it is not correct.
Here's the hint they give:
Write an equation for the total mass in the container in terms of the base area, height x of the candy in the container, mass of each candy, and volume of each candy. Then take a time derivative of each side of the equation.
When they say mass of the container in terms of base area -- do they mean LWH = m/roh??
Also, height of x -- is this the rate dH/dt?
The hint doesn't really help me -- it just confuses me more haha.
To find the rate at which the mass of the candies in the container is increasing, we can use the concept of density.
Density (ρ) is defined as mass per unit volume, expressed as ρ = m/V.
Here, we are given that the candies have a volume v and a mass m. Since the candies are identical, the density of each candy can be expressed as ρ = m/v.
Now, let's consider the container. The container has a base area of length L and width W, and the height of the candies is increasing at a rate of dH/dt.
The volume of the candies in the container is given by V = L * W * H, where H is the height of the candies.
To find the rate at which the mass of the candies is increasing, we need to differentiate the mass equation with respect to time:
d(m)/dt = d(ρ * V)/dt = ρ * d(V)/dt
Let's find d(V)/dt:
d(V)/dt = d(L * W * H)/dt = L * W * d(H)/dt
Substituting ρ = m/v and V = L * W * H into the equation for d(m)/dt:
d(m)/dt = (m/v) * (L * W * d(H)/dt)
So, the rate at which the mass of the candies in the container is increasing can be calculated by multiplying the density of each candy (m/v) by the rate at which the height of the candies is increasing (dH/dt).
Therefore, the rate at which the mass of the candies in the container is increasing is (m/v) * (L * W * d(H)/dt).