Question

Hello, do you guys know how to solve 2c-13-part b?
I cannot....

Answer 1

What we want to maximize is profit, and according to the statement of the problem we can consider it to be a function of price. Small p is used in one of the given equations, so let's not mess with that. So we will use M for profit.

We want a function M(p) -- profit as a function of price -- and then we want to maximize that. Now for the real-world, cultural part. What is profit? Well, capitalists and socialists may differ on where profit comes from, but I think there is general agreement that considering ourselves a vendor (we are an electric company) profit is sales minus costs. M(p) = S - C.

Both sales and cost are money amounts. Price p is a money amount. and M(p) profit is a money amount. This all seems promising. But the equations we have are in terms of units, which in this case are kilowatt hours. We do not really have to know how many kilowatt hours are involved, but we have to express them in terms of p -- the sale price.

How many units will be sold is given in terms of p: 10^5(10-p/2). This is units.

How much each unit costs is given for a number of units x: 10 - x/(10^5). This is money per unit.

We want both to be money (looking ahead) so units need to be multiplied by a money amount, and money per unit needs to be multiplied by units. That was just a flash forward.

M(p) = S - C.

So we want sales (in money) as a function of p (price). So pretty obviously, S(p) -- sales as a function of price -- units times price, buy we know the units, so

S(p) = 10^5(10-p/2)(p). Units times price = sales, a money amount.

M(p) = S(p) - C.

Now we know the cost per unit is 10-x/(10^5) where x the number of units. The cost is cost per unit times the number of units. But we know the number of units is 10^5(10-p/2)

so C(p) = 10 - (10^5(10-p/2)/(10^5)[10^5(10-p/2)]

And we have

M(p) = S(p) - C(p)

M(p) = 10^5(10-p/2)(p) - {10 - (10^5(10-p/2)/(10^5)[10^5(10-p/2)] }

whew, but 10^5/10^5 cancels, then factoring

M(p) = [10^5(10-p/2)][p - 10 + (10 - p/2)]

= [10^5(10-p/2)][p/2]

Multiplying two factors by 2 and dividing the other by 4

M(p)= [(10^5)/4] (20 - p) (p)

You should have no trouble differentiating in respect to p, finding the critical point, and checking that it is a maximum.

Notice the problem does not require finding what profit the utility makes or how many units it sells to maximize. The question is strictly what price it should charge.