Applications of Derivativs: 2 numbers w/ sum of 18, product is max; power lines

[bsandhu4]

Questions:

1. Find two positive numbers whose sum is 18 and the product of the first number and the square of the other is a maximum.

for this question i got the two numbers to be x=6 and y=12, is this right?

2. Two isolated are situated 12km apart on a straight country road that runs parallel to the main highways 20km away. The power decides to run a wire from highway to the junction box, and from there, wired of equal length to two houses. Where should the junction box be to MINIMIZE the length of wire needed?


I>-I
I

I represents the road
>-represents the junction
I represents the highway
I

Im very confused by this question.

Thanks for your help!!
I

 

[stapel]

1. Find two positive numbers whose sum is 18 and the product of the first number and the square of the other is a maximum.

for this question i got the two numbers to be x=6 and y=12, is this right?

No. Please reply showing your work, so that we can help you find your error(s).

2. Two isolated are situated 12km apart on a straight country road that runs parallel to the main highways 20km away.

"Two isolated" whats? "Houses", maybe?

The power decides to run a wire from highway to the junction box, and from there, wired of equal length to two houses.

By "the power", do you mean "the power company"? Should "wired" be "wires"?

Where should the junction box be to MINIMIZE the length of wire needed?


I>-I
I

I represents the road
>-represents the junction
I represents the highway
I
I

I'm sorry, but I don't know what the above is meant to say...?

Please reply with clarifications. It may be helpful to upload a scan of your drawing, or to use "code" tags to create a "drawing", like so:

drawing:

  *<----12--->*
      |
      |
      |
      |
      | 20
      |
      |
      |
      |
      |
---------hwy----

Please be complete. Thank you!

 

[ksdhart2]

1. Find two positive numbers whose sum is 18 and the product of the first number and the square of the other is a maximum.

for this question i got the two numbers to be x=6 and y=12, is this right?

No. Please reply showing your work, so that we can help you find your error(s).

Ehm... actually, Stapel, I think Bsandhu is correct on this one. If we designate the two unknown numbers as x and y respectively, then we're given that x + y = 18 with x > 0 and y > 0. Then the problem wants us to maximize "the product of the first number [x] and the square of the second number [y²]" so we'd need to maximize xy². This, indeed, has a local maximum at the point (6, 12). Granted, it's still theoretically possible for there to be one or more errors in OP's working, but the final answer is correct, no?

 

[stapel]

Ehm... actually, Stapel, I think Bsandhu is correct on this one. If we designate the two unknown numbers as x and y respectively, then we're given that x + y = 18 with x > 0 and y > 0. Then the problem wants us to maximize "the product of the first number [x] and the square of the second number [y²]" so we'd need to maximize xy². This, indeed, has a local maximum at the point (6, 12). Granted, it's still theoretically possible for there to be one or more errors in OP's working, but the final answer is correct, no?

I missed "the square of". Oops!

Thank you for catching that!