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You own a hamburger stand. Your specialty is delicious cheeseburger that you sell for USD $7.00$ each. The cost of ingredients for each cheeseburger, such as meat, cheese, buns, tomato etc., is USD $1.65$. You hire workers at USD $12$ an hour. The production function for cheeseburgers is $Q = 40 \sqrt{L}$ , where L is the number of labor hours employed and $Q$ the number of cheeseburgers produced. There is a fixed cost of USD $172$.

There is no way to produce delicious cheeseburgers without also producing lots of grease, which needs to be hauled away. In fact, every cheeseburger that you produce, you also produce 1 ounce of grease. Lately, cooking grease has become popular as an ingredient to produce alternative fuel. So, a company begins to buy grease from you every day at the rate of USD $0.80$ per pound, thereby eliminating your cost of hauling away grease and providing you another revenue source.

Calculate the daily profit maximizing number of hamburgers, the amount of grease produced, the number of labor hours used and the consequent level of profit.

I have actually come up with an equation and differentiated it to get an answer. Just want to be sure it is right. My equation : $(-3x^2 + 2140x - 68800) / 400x$.

Number of burgers for maximum profit are coming out to be $151/152$. Does this sound right or am I going wrong somewhere?

Henry
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Tyler Fish
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3 Answers3

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Hint:

Suppose you want to produce and sell $Q$ hamburgers a day

  • On the cost side
    • How many working hours do you need? How much does that cost?
    • How much do the ingredients cost?
    • What is the fixed cost?
    • What is the total cost?
  • On the income side
    • How much do you earn from the sales of hamburgers?
    • How much do you earn from the sales of grease? ($16$ ounces in a pound)
    • What is the total income?
  • Putting these together
    • What is your profit from $Q$ hamburgers?
    • Which $Q$ maximises your profit?
    • How much grease corresponds to this optimal $Q$?
    • How many working hours correspond to this optimal $Q$?
    • How much profit corresponds to this optimal $Q$?
Henry
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  • Thank you for the response. I have actually come up with an equation and differentiated it to get an answer. Just want to be sure it is right.

    My equation : (-3x^2 + 2140x - 68800) / 400x Number of burgers for maximum profit are coming out to be 151/152.

    Does this sound right or am I going wrong somewhere?

     – Tyler Fish Nov 25 '20 at 10:55
  • Also, I have kept the grease part separate (not a part of the equation above), because that did not feel like making a difference to the burger profit part. (Also hoping I did not do a mistake in doing this, please correct me if I'm wrong) – Tyler Fish Nov 25 '20 at 10:58
  • I do not see how you get your formula. Why are you dividing by $x$? You do not care about profit per burger but overall profit. Why by $400$ rather than something else? And yes you are supposed to take account of the grease (I think it makes a very small difference but leads to an easy integer result, much smaller than yours) – Henry Nov 25 '20 at 11:16
  • Yea, sorry I was looking at it as profit per burger. – Tyler Fish Nov 25 '20 at 11:25
  • Okay I changed it to overall profit, and the equation came the same as the 2nd answer somebody posted below but in Q, I don't know why they left it in L terms.

    Final Equation: -3/400Q^2 - 27/5Q - 172 Maximum Q = 360 now.

     – Tyler Fish Nov 25 '20 at 11:35
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Total cost of ingredients $a$ = $1.65Q = 66 \sqrt L$

Labor cost $b$ = $12L$

Fixed cost $c = 172$

Profit $ = 280 \sqrt L - (66 \sqrt L + 12L + 172) = 214 \sqrt L - 12 L - 172$

Grease produced = $Q$ = $40\sqrt L \,$ ounces = $2.5 \sqrt L$ pounds.

Profit from grease $ = 0.80 \times 2.5 \sqrt L = 2 \sqrt L$

Total profit $P = 216 \sqrt L - 12 L - 172$

EDIT: expressing in terms of $Q$,

$P = \frac{216}{40} Q - \frac{12}{1600} Q^2 - 172$

i.e. $\, P = \frac{27}{5} Q - \frac{3}{400} Q^2 - 172$

Math Lover
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Math Lover and Henry have given nice answers to the OP's cheeseburger problem; this answer addresses the OP's italicized question, Does this sound right or am I going wrong somewhere?

One mistake is in ignoring the contribution of the sale of grease, so the $2140$ in the expression $(-3x^2+2140x-68800)/400x$ should really be $2160$.

But the real problem is that the expression $(-3x^2+2140x-68800)/400x$ represents the profit per cheeseburger, which is not what it is you want to maximize. Instead, you want to maximize the total profit, which is simply $(-3x^2+2140x-68800)/400$ (or, rather, $(-3x^2+2160x-68800)/400$, correcting for the omission of the grease contribution). It may, superficially, sound like these are the same two things to maximize, but they aren't, and it'll be time well spent to think about why they're not the same.

Finally, two hopefully helpful suggestions. First, it's OK to use $x$ as your variable, but it would be easier to stick with $Q$; at the very least you should somewhere or other explicitly say what $x$ represents. And second, $151/152$ looks like a fraction, slightly less than $1$, which seems to suggest you've concluded the profit maximization occurs by making (and selling) something short of a single cheeseburger, which makes little sense. It took me a while to realize that you meant the profit (per burger) is maximized at a value of $x$ between $151$ and $152$ (i.e., $x=\sqrt{68800/3}\approx151.43755$). In short, please don't make us guess at the meaning of your expressions!

Barry Cipra
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