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The lotal-cost function for an electric lighl and power plant Is estimaled lo be c=jZ 07 = 0 79q 02142q" 0 000 IIq" 104q = 100 Where I5 the eighl-hour tol...

Question

The lotal-cost function for an electric lighl and power plant Is estimaled lo be c=jZ 07 = 0 79q 02142q" 0 000 IIq" 104q = 100 Where I5 the eighl-hour tolal outpul (as percenlage 0i capacih ) and I5 Ihe Iotal fuel cost in dollars Find Ihe marginal-cost funclion and evaluate i when q= 48 Desermineshe margial-cost function Select the correct chaice below and fill in the answer box to complete your choice (Use integers Or decimals for any numbers in the expression /0 8

The lotal-cost function for an electric lighl and power plant Is estimaled lo be c=jZ 07 = 0 79q 02142q" 0 000 IIq" 104q = 100 Where I5 the eighl-hour tolal outpul (as percenlage 0i capacih ) and I5 Ihe Iotal fuel cost in dollars Find Ihe marginal-cost funclion and evaluate i when q= 48 Desermineshe margial-cost function Select the correct chaice below and fill in the answer box to complete your choice (Use integers Or decimals for any numbers in the expression / 0 8



Answers

The monthly demand equation for an electric utility company is estimated to be $$p=60-\left(10^{-5}\right) x,$$
where $p$ is measured in dollars and $x$ is measured in thousands of kilowatt-hours. The utility has fixed costs of 7 million dollars per month and variable costs of $\$ 30$ per 1000 kilowatt-hours of electricity generated, so the cost function is
$$a(x)=7 \cdot 10^{6}+30 x,$$
(a) Find the value of $x$ and the corresponding price for 1000 kilowatt-hours that maximize the utility's profit.
(b) Suppose that rising fuel costs increase the utility's variable costs from $\$ 30$ to $\$ 40$, so its new cost function is
$$C_{1}(x)=7 \cdot 10^{6}+40 x.$$
Should the utility pass all this increase of $\$ 10$ per thousand kilowatt-hours on to consumers? Explain your answer.

And this problem, you're asked the cost for each vehicle of driving 12,000 miles per year. Now, in the previous problem, we found what a is based on this table here and what B is based on this table here and we multiplied and being Do you get this matrix here now to get the cost of driving each car? It's actually quite simple. All we do is take 12,000 times the matrix eight times being, and this will give us the total cost because we're essentially taking 12,000 miles times a cost her a mile in the miles. Cancel out to just give us the cost. And so when we do this math, we get the following. We get 2736 972 and 180. Now. This tells us that the cost of driving an SUV per year if you go 12,000 miles is $2736. The cost of driving a hybrid car is $972. In the cost of driving, an electric car is $180

A few different parties fear. Let's get right to it was the increase in average cost per widget where we can see $4 55th 55 cents per or $4.90 8 since Percy do well subtraction and you will see what that differences, which it looks like is 43 cents. So 43 cents is the difference in cost per widget from that kind of increase in price. And so before. So we're looking at that first equation. They were sold for $8 each, which means we take our equation. Do it here. Awesome rooms. Our equation. Let me. Actually, I'm not gonna write. You'll see why in a bit, um $4.554.55 since per widget for 69,000. This is the amount of money spent. So the break even point will be if I spend $8 per widget and multiply that by the quantity, this is the amount earned, and all this over here is the amount spent So we need to find when those two things are equal. So now we do algebra. Let's subtract 4.55 drum or a Q 4.55 Q from both sides. The equation. Eliminating it over here and isolating the Q over there, leaving us with 3.45 Q equal to 69,000. One more step Mr Abide by 3.45 on both sides of the equation. And if I divide 69,000 by 3.45 I kind of ran out of room. But I'll just say this. The quantity is 20,000 now. That's actually not just it's all right. First of all, that answer here said Super, super clear. Um, if it was $8 each, you actually have to also multiply that times eight. So 20,000 widgets, or if I multiply that by 8 160,000 dollars turned. So those are both really part of that answer, All right, pretty much the exact same question. Except it says Okay, the increased the price to 8 50 and now we're using different equation. It's let me scroll down. So we have enough from 4.98 just to remember. Okay, so now we have 4.98 tons of quantity. Same fixed costs of $69,000. Now. This time it's set equal to 8.5. Most tried by the quantity. Same for step when we scroll down again. So, yeah, I even scroll down. So that's out of the frame there. Same for step. We want to put all the cues on one side of the equation, and that's done by subtracting four 0.9 a Que and let me double check what that leaves us with for 8.5 minus 4.9 A. Just don't make sure I don't make a silly mistake is 3.52 que you cool to 69,000 step divide by over 13.52 on both sides of the equation, leaving us just with Q over here. And I'm pretty sure this is not a whole number. Which is why they told us around to the nearest whole, surrounded to the nearest Wallace. 19,602 widgets. Of course, you have tow. Multiply that by the price, so that's part of the answer. That's a quantity. But if I multiplied by the price of $8.50 that's gonna give me what that break even amount of money. ISS some 19 c 19,006 or two. I want to make sure don't make a mistake. Is 100 and $66,617. There we go. So me kind of have I and Green that 19,602 widgets or 166 1006 $117.

In this problem here. Given these two tables and asked to write Matrix A matrix represents the consumption in matrix B were major speed ever since the costs. Now we're gonna write Matrix A by using the numbers in this table here, and we're gonna write Matrix being by using the numbers in this table here. And so Matrix A is going to be a three by two matrix, as it tells us, and it's going to be a 20.6 zero 0.2 0.1 0.3 In matrix being is going to be a two by one. Matrix is going to be 3.80 and point 05 Okay, so in l. A, we found our major sees A and B, you were going to find what a Times B is equal to. No, again A is a three by two matrix and B is a two by one matrix, so we can multiply them because the middle numbers are equal and our final matrix is going to be three by one. And so I'm going to right that here and to multiply, these are going to look at the less matrix for rose and the right matrix for column. So first, I'm going to take 0.6 times 3.80 zero times 2.5 and I get point to to a Then to get our second component, we're going to take 0.2 times 3.80 plus 0.1 times 0.5 and we get 0.81 Lastly, to get the third component, we're going to take zero times 3.80 plus 0.3 times 0.5 and we get zero. Excuse me. 0.1 thoughts. So we've found our majors, a B, and now we're asked to say what the Matrix represents. Now, what we just did is now what we just did it was multiply the number of either gallons or kilowatt hours, huh? Smile times the cost. Her gallon or kilowatt hour. Now, as you can see, the gallons and kilo hours cancel out and we're left with cost her mind. And so our matrix A b represents it cost her mile for each vehicle. That is our answer. So that tells us that for the SUV, the cost per mile is point 2 to $8 they cost per mile for the hybrid car is 0.1 dollars per mile, and they cost for the electric car is $0.15 per mile.

Okay and electric company charges its customers a base rate of ten dollars a month plus six cents per kilowatt hour for the first twelve hundred kilowatts and seven cents per kilowatt hours for all Houston's over twelve hundred kilowatt hours. Express the monthly cost he as a function of the amount x of electricity used then Graff the function e for exes between zero in two thousand. Okay, so for this cost function G of X for when we use less than less than twelve hundred kilowatts, I guess we have a base rate of ten plus six cents for each kill a lot we use Okay, And then what if we use more than twelve hundred? Cure alliance. Okay, so I guess we'll have ten dollars for the base tree class. Six cents times twelve hundred. Which is gonna be Let me put that into my calculator really quick. You know your point zero six times twelve hundred, which is seventy two. So we have eighty two plus this seven cents Hercules times the amount of kill lost of use and that will be represented by X minus twelve hundred. This will be our cost function for the amount of electricity that we use okay. And to graft this maybe we'll just do a rough sketch here. You didn't color or he that of the ex. I guess we have a base rate of ten until we get twelve hundred kilowatts. So that'LL be a positive slope here, oddly enough there. And then now that's a slope of zero point zero six. Then now, after twelve hundred will have slope, that's slightly more than zero point zero six say zero point zero seven. So we'LL need to have that slightly a steeper This is just a rough sketch and then the always two thousand kilowatts.


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