Optimization or Related Rates to solve this problem?

Question

The problem: A company has $120 000 to spend on the development and promotion of a new product. The company estimates that if x is spent on the development and y is spent on promotion, then approximately (x^(1/2)y^(3/2))/(400000) items of new product will be sold. Based on this estimate, what is the maximum number of products that the company can sell?

Not sure if this is an optimization or related rates problem, but even then I am not sure as to how to start it. I know the answer is supposed to be 11691.

chris · Answer 1 · 2019-12-08T03:27:30.960

0

The critical observation here is to notice that y = 120000-x, so your expression simplifies to:

On the plane,

The max occurs at x=30000 (you can verify analytically by finding zeroes of the first derivative); substituting, you'll find the max number of products to be 11,691.

edited Dec 08 '19 at 03:27

answered Dec 08 '19 at 03:10

chris

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I did manage to make y = 120000 - x, but failed to see how it fit into the equation. Right in front of my eyes, I appreciate the clarification and guidance. – boomboomboth21 Dec 08 '19 at 03:27
No problem, glad to help. Welcome to SO! – chris Dec 08 '19 at 03:45

Optimization or Related Rates to solve this problem?

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