Question

I have the answers for this problem but I don't understand how to get the answer:

Suppose that x represents one of the two positive numbers whose sum is 30.

a) Represent the other two numbers in terms of x
b) What restrictions are on x
c) Determine a function P that represents the product of two numbers
d) Determine analytically and support graphically the two such numbers whose product is a maximum, what is the maximum product.

Answer 1

Two numbers add up to 30.
One of the numbers is x
So the other number must be (30 - x)
So the two numbers are: x and (30 - x) (and you can verify they do indeed add up to 30)

Answer 2

b) Restrictions on x? The problem states x is a positive number. So x > 0 is the restriction

c) Product of the two numbers P(x) = x(30 - x)

d) Try various values of x and find the product
p(1) = ?
p(2) = ?
p(3) = ?
Can you fill in the question marks? Then I will help you find the maximum

Answer 3

29
56
81

Answer 4

Correct. Keep doing until you reach x = 15. You have already done for x = 1, 2 and 3.

Answer 5

I don't understand, why we would only do it up to x=15

Answer 6

First do till 15 and then I was going to ask you to do just two more 16 and 17 and you will be able to figure out what is happening.

So list P(x) for x = 1, 2, 3, 4, ..... 15, 16, 17

Answer 7

oh I see then 15 will be the maximum because aftter 15 the number decreases

Answer 8

Exactly!

Answer 9

So the answer to d) is P(15) = (15)(15) = 225.
Then you have to graph it.

Answer 10

is there another way I could solve the maximum without going thru all the numbers

Answer 11

The problem states do it analytically and graphically.
I don't know if you have already been introduced to calculus. Have you been?

Answer 12

If you know calculus then simply solve for dP(x)/dx = 0 and you will know what value of x maximizes p(x).

Answer 13

no I haven't, I'm a current college freshman and I had horrible teachers my junior and senior year of high school so everything is new to me

Answer 14

With calculus you can find the maximum in a couple of steps.
But since they expect to you to graph P(x) it will be useful to know what P(x) is for various values of x.
The graph will be an inverted parabola initially increasing until until x = 15 and then it will start to decrease.

Answer 15

oh okay than

Answer 16

*
thank you

Answer 17

u r welcome.