Find the number whose sum is 15. If the product of the square of one by the cube of the other is to be maximum
a+b=15
a^2b^3=m
a=15-b
(15-b)^2b^3=m
(225-30b+b^2)b^3=m
225b^3-30b^4+b^5=m
dm/db=675b^2-120b^3+5b^4
at dm/db=0
675b^2-120b^3+5b^4=0
b^2(675-120b+5b^2)=0
divide through by b^2
5b^2-120b+675=0
divide through by 5
b^2-24b+135=0
plz solve quadracticly and sub into the first equation to find a
b^2-24b+135 = (b-9)(b-15)
So, the extrema are at b=9 or b=15.
Naturally, at b=15, a=0, so the product is a minimum
At b=9, a=6, and the product is
6^2*9^3 = 26244
a+b=15
a^2b^3=m
To find two numbers whose sum is 15 and whose product of the square of one by the cube of the other is to be maximum, we can use a mathematical approach.
Let's assume one of the numbers is "x", and the other number is "15 - x" (since the sum of the two numbers is 15).
To maximize the product, we need to maximize the value of (x^2) * ((15 - x)^3).
To find the value of "x" that maximizes the product, we can use calculus. In this case, we need to find the derivative of the product function with respect to "x" and find the critical points.
Let's calculate the derivative:
f(x) = (x^2) * ((15 - x)^3)
f'(x) = 2x * ((15 - x)^3) + (x^2) * 3 * ((15 - x)^2) * (-1)
To find critical points, we need to make f'(x) equal to zero and solve for "x":
2x * ((15 - x)^3) + (x^2) * 3 * ((15 - x)^2) * (-1) = 0
Simplifying the equation:
2x * ((15 - x)^3) = 3x^2 * ((15 - x)^2)
Dividing both sides by "x" (assuming x ≠ 0):
2 * ((15 - x)^3) = 3x * (15 - x)^2
Expanding the equation:
2 * (3375 - 675x + 45x^2 - x^3) = 3 * (225x - 15x^2 + x^3)
Simplifying further:
6750 - 1350x + 90x^2 - 2x^3 = 675x - 45x^2 + 3x^3
Rearranging the equation:
3x^3 + 2x^3 - 45x^2 + 90x^2 - 675x + 1350x - 2x^3 - 6750 = 0
Now we can combine like terms:
5x^3 + 45x^2 + 675x - 6750 = 0
At this point, we can use numerical methods or software to find the solutions for "x". The values of "x" that satisfy this equation will be critical points of the product function.
Once we have the critical points, we evaluate the function f(x) = (x^2) * ((15 - x)^3) at these values and select the maximum value that corresponds to maximum product.
However, if you just need the value of the numbers whose sum is 15 and whose product (square of one by the cube of the other) is maximum, we can provide it directly without the need for calculus:
Since the maximum product is desired, we can set the two numbers as close as possible to each other. In this case, we can set both numbers as 7.5, where one number is slightly less than 7.5 and the other is slightly greater than 7.5.
Therefore, the numbers whose sum is 15 and whose product of the square of one by the cube of the other is to be maximum are approximately 7.5 and 7.5.