I need math help. I don't want to cheat. Please show steps so I can learn. Thanks. Much appreciated.?

Question

The sum of one number and two times another number is 24. Find the two numbers such that their product is a maximum.

(I know that the answer are 12 and 6, please show me the equation to get these answers) thx

Answer 1

Let the two numbers be x and y.

Then the first statement says:
x + 2y = 24

We want to maximise xy. Since x = 24 - 2y we can recast this as a function of y alone:

f(y) = xy = (24 - 2y) . y = 24y - 2y^2.

(We could also write it as g(x) = x (12 - x/2), but I prefer to keep integer coefficients - especially in text!)

Take the derivative of f(y) and you get 24 - 4y which is zero at y = 6. The second derivative is -4 so this is a maximum of the function. Finally, use x = 24 - 2y to get x = 12; so the maximum value occurs when the first number mentioned is 12 and the second number is 6.

Answer 2

Let's assume that x is the first number and y is the second, then:
x + 2y = 24
When we want their product to be maximum, then this means that the first derivative of their product should equal 0, then assuming that the product is z, then:
z = x*y= (24-2y)*y = 24y-2y^2
deriving z:
z' = 24-4y
To get the maximum, then z' should equal 0
0=24-4y, which means that y=6
Solving for x, we should substitute in the first equation:
x+2y= 24, then x= 24-2y= 24-12=12

Hope this was clear enough

Answer 3

Here y+2x=24 or y=24-2x, z=x*y or z=(24-2x)*x; here comes z’(x)=24-4x=0, hence x=6 and y=12

Answer 4

Let nos be x and y

x+2y=24

product=xy

x=24-2y

prod p=y(24-2y)=24y-2y^2

dp/dy=0

24-4y=0

y=6

for prod to be maximum,d2p/dy2<0

d2p/dy2=-4

so y=6 s correct soln

x=12

Answer 5

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Answer 6

let x = 1st #
let 2x = 2nd #

x + 2x = 24

3x = 24

x = 24/3

x = 8

therfore, x = 8 and 2x = 16

check:

8 + 16 = 24.

hope i can help!!!