This problem is a related rates problem with two related rat

*This problem is a related rates problem with two related rates. Please attempt because I am stuck*

A company will be introducing a new sandwich cookie with orange creme filling. After the cookies are made, they will be funneled single­ file onto a conveyer belt and what we’re calling the “squirt machine” will add a squirt of filling to each cookie. Later a top cookie is added and the cookies move on to packaging, but my responsibility has to do with this filling phase. In short, I need to figure out the appropriate speed for the conveyer belt and program this into the system. We need to make sure that we don’t end up with any creme­less cookies (which unfortunately happened during our first trial run.) The creme filling is made in a machine that has the shape of a box with a triangular prism (or trough) on the bottom.

-The machine is 35 in tall, 25 in wide and 16 in deep, and the top box part is 25 in high, 25 in wide, and 16 in deep. The sides of the prism are isosceles triangles.

-It has some sort of plunger device that pushes the orange creme filling from the container and into the connection leading to the squirt machine. The plunger is a horizontal plate which starts at the top of the fully ­filled machine and lowers at a constant rate of 0.15 in/min.

- The fully ­filled creme filling machine has a volume of 12,000 in3. Each “squirt” of filling has a volume of 0.1 in3. The diameter of the circular ­shaped cookies is 1 in. The maximum speed of the conveyer belt is 980 in/min.

1)Use related rates to figure out the speed that the conveyer belt should run.

Solution

Here, I will make a few assumptions before solving, as I don\'t see any mention about them in the problem statement. However, I will keep the solution in terms of the variables until last subsitution so that you can edit and proceed.

First, the cookies are right next to each other, as I would need the time gap between each squirts. [If it is otherwise, you can simply edit the distance \'d\' between the cookies]

Second, the plunger travels till the bottom end of the filler as in till the bottom of the prism, and doesn\'t stop at the end of the cuboidal section.

Third, one single \'squirt\' will be enough for the cookie and is expected to be dropped at the centre of it.

Now, as the plunger moves down at a fixed constant rate, the time which we have to squirt the entire amount out will be controlled by the plunger which in turn will determine the optimum speed for the conveyor.

We will call the vertical distance which the plunger has to travel the given rate of 0.15 in/min as H inches.

So the time we have to empty the machine is (H / 0.15) minutes

Also, 12000 in^2 of cream is to be pushed out in squirts of 0.1 in² each, which would mean that a total of 12 x 10⁴ squirts are to be made in (H/0.15) minutes

That would mean that 12 x 10^4 cookies of diameter 1 inch must pass across the squirt machine in the given time. So the belt must travel a distance of 12 x 10^4 inches in the time (H/0.15) minutes.

Therfore the speed of the conveyor is given as:

S = 12 x 10^4 x 0.15 / H = 0.0514286 x 10^4 inches/min = 514.286 in/min is the required speed of the conveyor

NOTE: As mentioned at the top, the value of H, it at all it is not supposed to be the entire height, can be subtituted and rate can be determined.