A vertical container with base area measuring 12 cm by 18 cm is being filled with identical pieces of candy, each with a volume of 50.0 mm3 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 0.29 cm/s, at what rate (kilograms per minute) does the mass of the candies in the container increase?

Question

A vertical container with base area measuring 12 cm by 18 cm is being filled with identical pieces of candy, each with a volume of 50.0 mm3 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 0.29 cm/s, at what rate (kilograms per minute) does the mass of the candies in the container increase?

Answer 1

Fullerton is a city in Orange county, California. There is a Cal State campus there. It is not the subject of your question.

The product of the area of the base area and the 0.29 cm/s rate of height increase equals the cm^3/s candy addition rate, which you should solve for.

Multiply it by the candy density for the mass addition rate.

Answer 2

To find the rate at which the mass of the candies in the container increases, we need to know the rate at which the volume of the candies in the container increases.

Given:
- The base area of the container is 12 cm by 18 cm.
- Each candy has a volume of 50.0 mm^3.

To find the height at which the candies fill the container, we can use the formula:

Volume of candies = Base area × Height.

Let's calculate the height of the candies:

Volume of candies = 12 cm × 18 cm × Height
50.0 mm^3 = 12 cm × 18 cm × Height

Now, we can find the rate at which the height of the candies increases by differentiating with respect to time:

d(Volume of candies)/dt = (d(Base area)/dt) × Height + (d(Height)/dt) × (Base area)

Here, we know that the rate at which the base area changes is zero because it is not changing. So, the equation simplifies to:

d(Volume of candies)/dt = (d(Height)/dt) × (Base area)

Now, we can substitute the given values into the equation:

0.02 g = (d(Height)/dt) × (12 cm × 18 cm)

To find the rate at which the mass of the candies in the container increases, we need to find the derivative of the height with respect to time:

d(Height)/dt = 0.29 cm/s

Substituting this value into the equation:

0.02 g = 0.29 cm/s × (12 cm × 18 cm)

Now, we can convert the units to kilograms per minute.

1 g = 0.001 kg
1 cm = 0.01 m
1 s = 60 min

Converting the units:

0.02 g = (0.29 cm/s × 12 cm × 18 cm) × 0.001 kg/g × 0.01 m/cm × 60 s/min

Simplifying the equation:

0.02 g = 0.00396 kg/min

Therefore, the rate at which the mass of the candies in the container is increasing is 0.00396 kg/min.