Question

find the two numbers whose sum is 12, if the product of one by the square of the other is to be maximum

a. 6 and 6
b. 5 and 7
c. 4 and 8
d. 9 and 3

Answer 1

We'll note the numbers as x and y.

The sum of the nubmbers is 12.

x + y = 12

y = 12 - x

We also know that the product of one and the square of the other is a maximum.

We'll write the product of numbers as:

P = x*y^2

We'll substitute y by (12-x) and we'll create the function p(x):

p(x) = x*(12-x)^2

We'll expand the square:

p(x) = x(144 - 24x + x^2)

We'll remove the brackets:

p(x) = 144x - 24x^2 + x^3

The function p(x) is a maximum when x is critical, that means that p'(x) = 0

We'll calculate the first derivative for p(x):

p'(x) = (144x - 24x^2 + x^3)'

p'(x) = 144 - 48x + 3x^2

p'(x) = 0

144 - 48x + 3x^2 = 0

We'll divide by 3 and we'll apply symmetric property:

x^2 - 16x + 48 = 0

We'll apply the quadratic formula:

x1 = [16+sqrt(256-192)]/2

x1 = (16+8)/2

x1 = 12

x2 = (16-8)/2

x2 = 4

The positive numbers are: x = 4 and y = 12 - 4 = 8

Answer 2

which is c .

Answer 3

wow u got it and nice explaination

Answer 4

is it easier to substitute x?

x + y = 12

xy^2 = p(x)

(12 - y)y^2 = p(x)

p'(x) = -y^2 + 2y(12 - y^2)

-3y^2 + 24y = 0

y(24 - 3y) = 0

y = 8

x = 4

Answer 5

w8 9 and 3 is also like that..

Answer 6

3*81 = 243
4*64 = 256

Answer 7

yes8 and 4

Answer 8

ok he says 8 and 4

Answer 9

u want the maxim, so 3*9^2=243 ,while 4*4^3=256

Answer 10

plus we just proved that with maths

Answer 11

the maximum in all letters is c =256 if i get the minimum the answer is the lower 1?

Answer 12

i fyou said for minimum product, the answer would be 6, 6

Answer 13

ty raza

Answer 14

U WLCM