A vertical container with base area measuring 12.3 cm by 13.7 cm is being filled with identical pieces of candy, each with a volume of 50.0 mm^3 and a mass of 0.0200 g. Assume that the volume of the empty spaces between the candies is negligible. If the height of the candies in the container increases at the rate of 0.306 cm/s, at what rate (in kg/min) does the mass of the candies in the container increase?

Question

A vertical container with base area measuring 12.3 cm by 13.7 cm is being filled with identical pieces of candy, each with a volume of 50.0 mm^3 and a mass of 0.0200 g. Assume that the volume of the empty spaces between the candies is negligible. If the height of the candies in the container increases at the rate of 0.306 cm/s, at what rate (in kg/min) does the mass of the candies in the container increase?

Answer 1

rate of mass change:

.360cm/s*(12.3*13.7)cm^2 * .02g/50mm^3 * (10mm/1cm)^3 * 1kg/1000g* 60sec/min

Put this in your google search window:
.360*12.3*13.7*.02/50*10^3/1000*60=

Answer 2

To find the rate at which the mass of the candies in the container is increasing, we need to calculate the rate of change of mass with respect to time. We can do this by using the chain rule from calculus.

Let's first find the volume of the candies in the container. The volume of each candy is 50.0 mm^3, and the base area of the container is 12.3 cm * 13.7 cm = 168.51 cm^2. Since the height of the candies is increasing at a rate of 0.306 cm/s, the volume of the candies in the container is increasing at the same rate, which is 0.306 cm^3/s.

To find the rate at which the mass is increasing, we need to find the rate of change of mass with respect to volume. We know that the mass of each candy is 0.0200 g. So the rate of change of mass with respect to volume (dm/dV) is simply the mass of each candy, which is 0.0200 g / 50.0 mm^3.

Now we have the rate of change of volume with respect to time (dV/dt = 0.306 cm^3/s) and the rate of change of mass with respect to volume (dm/dV = 0.0200 g / 50.0 mm^3). To find the rate at which the mass is increasing with respect to time (dm/dt), we can use the chain rule:

dm/dt = (dm/dV) * (dV/dt)

Substituting the values we know:

dm/dt = (0.0200 g / 50.0 mm^3) * (0.306 cm^3/s)

To convert the units to kg/min, we need to make the following conversions:
- Convert mm^3 to cm^3: 1 mm^3 = 0.001 cm^3
- Convert g to kg: 1 g = 0.001 kg
- Convert s to min: 1 min = 60 s

So, the final calculation is:

dm/dt = (0.0200 g / 50.0 mm^3) * (0.306 cm^3/s) * (0.001 kg / 1 g) * (1 min / 60 s)

After performing the calculations, we find that the rate at which the mass of the candies in the container is increasing is approximately 0.000002 kg/min.