Duke Energy Calculus Project by

Optimization Problem

tiduy96 — 2013-11-17 18:36:45 UTC

Minimize the total cost of underground wiring

Problem

tiduy96 — 2013-11-17 19:14:58 UTC

- The power line needs to run from the transformer (the green dot) to the light (the red dot) .

- The blue path will be done by the trench-digging device, which costs $15/ft while the orange path will be done by the horizontal boring machine at $40/ft.

- Determine the location where the horizontal boring machine should be placed to minimize cost.

Solution

tiduy96 — 2013-11-17 19:21:58 UTC

tiduy96 — 2013-11-17 19:31:45 UTC

Find the derivative

tiduy96 — 2013-11-17 19:39:25 UTC

There is a local min at x = 60.678 since C' changes from negative to positive

tiduy96 — 2013-11-17 19:51:30 UTC

The distance down the cart path for minimum cost:

tiduy96 — 2013-11-17 19:59:48 UTC

The total real power from the Shearon-Harris nuclear plant and the Roxboro coal plant

choheo99 — 2013-11-17 20:14:35 UTC

Problem

tiduy96 — 2013-11-17 20:18:40 UTC

Find the rate at which the power factor is changing when the reactive power is increasing at a rate of 300 MVAR/hr and the power factor is 0.99.

Solution

tiduy96 — 2013-11-17 20:21:45 UTC

Power Triangle

tiduy96 — 2013-11-17 20:27:10 UTC

Formula for power factor

tiduy96 — 2013-11-17 20:28:41 UTC

To find c

tiduy96 — 2013-11-17 20:38:13 UTC

Using Pythagorean Theorem, we can find a as we know b and c

tiduy96 — 2013-11-17 20:55:39 UTC

And now the rate of change of c

tiduy96 — 2013-11-17 21:01:09 UTC

Now that we have c and the rate of change of c, we can find the rate of change of the power factor!

tiduy96 — 2013-11-17 21:01:28 UTC

tiduy96 — 2013-11-17 21:08:07 UTC

Write a function to represent the total cost and simplify

tiduy96 — 2013-11-18 00:58:42 UTC

Set the derivative to 0 to find local extremas

tiduy96 — 2013-11-18 01:05:48 UTC

tiduy96 — 2013-11-18 01:11:26 UTC

Derive the power factor equation explicitly

tiduy96 — 2013-11-18 01:26:26 UTC

So...

tiduy96 — 2013-11-18 01:31:44 UTC

We need to find c (or apparent power) and the rate of change of c to get the answer

To find the rate of change of c

tiduy96 — 2013-11-18 01:36:28 UTC

First we need to find a

Note

tiduy96 — 2013-11-18 01:58:30 UTC

The real power b is a constant, so its derivative is 0.

The total minimum cost is:

tiduy96 — 2013-11-18 02:08:31 UTC

The power factor is decreasing at a rate of 0.01249 per hour at 10 am when the rate of change of reactive power is 300 MVAR/hr and the power factor is 0.99

tiduy96 — 2013-11-18 02:23:00 UTC

Duke Energy Calculus Project by

Optimization Problem

Related Rates Problem

Problem

Solution

Find the derivative

There is a local min at x = 60.678 since C' changes from negative to positive

The distance down the cart path for minimum cost:

The total real power from the Shearon-Harris nuclear plant and the Roxboro coal plant

Problem

Solution

Power Triangle

Formula for power factor

To find c

Using Pythagorean Theorem, we can find a as we know b and c

And now the rate of change of c

Now that we have c and the rate of change of c, we can find the rate of change of the power factor!

Write a function to represent the total cost and simplify

Set the derivative to 0 to find local extremas

Derive the power factor equation explicitly

So...

To find the rate of change of c

Note

The total minimum cost is:

The power factor is decreasing at a rate of 0.01249 per hour at 10 am when the rate of change of reactive power is 300 MVAR/hr and the power factor is 0.99