divide 64 into two parts such that the product of one of them plus the cube of the other is a maximum
x+y=64
z = xy^3 = (64-y)y^3 = 64y^3 - y^4
dz/dy = 192y^2 - 4y^3 = 4y^2(48-y)
I think the way is now clear, right?
A product is the result of a multiplication.
"product of one of them plus the cube of the other" is mathematically confusing, since "plus" implies addition.
I will assume you meant:
"product of one of them with the cube of the other"
let the one to be cubed be x
then the other is 65-x
let S be the product of one of them plus the cube of the other
S = x^3(65-x) = 65x^3 - x^4
dS/dx = 195x^2 - 4x^3
= 0 for a max of S
195x^2 - 4x^3 = 0
x^2(195 - 4x) = 0
x^2 = 0 ---> x = 0
then the product as stated would be zero, not very big
or
4x = 195
x = 195/4
one is 195/4 , the other is 65 -195/4 or 65/4
as decimals:
48.75 and 16.25
dang those typos, yeah? 64 works much better.
dont' know where the 65 came from
simply change it to 64 and complete the solution,
you should get 48 and 16
To solve this problem, we need to find two numbers that add up to 64 and maximize the product of one number and the cube of the other number.
Let's assume the first number is x, and the second number is 64 - x.
We need to maximize the product of x and the cube of (64 - x). So, our objective function is f(x) = x * (64 - x)^3.
To find the maximum value of f(x), we can proceed as follows:
1. Take the derivative of f(x) with respect to x.
2. Set the derivative equal to zero and solve for x.
3. Check the second derivative to determine whether the critical point is a maximum or minimum.
4. Plug the critical point into f(x) to find the maximum value.
Step 1: Taking the derivative of f(x):
f'(x) = (64 - x)^3 - 3x(64 - x)^2
Step 2: Set f'(x) = 0 and solve for x:
(64 - x)^3 - 3x(64 - x)^2 = 0
Expand and simplify:
64^3 - 3*64^2x + 3*64x^2 - x^3 - 3*64^2x + 6*64x^2 - 3x^3 = 0
Combine like terms:
64^3 - 6*64^2x + 9*64x^2 - 4x^3 = 0
Factor out common terms:
64(64^2 - 6*64x + 9x^2) - 4x^3 = 0
Divide both sides by 4:
16(64^2 - 6*64x + 9x^2) - x^3 = 0
Step 3: Checking the second derivative:
f''(x) = -6(64^2 - 6*64x + 9x^2) -3x^2
To maximize the function, we need the second derivative to be negative.
Step 4: Plug the critical point into f(x) to find the maximum value.
Substitute the critical point value of x back into our objective function f(x) = x * (64 - x)^3 to find the maximum value.
Unfortunately, solving this equation analytically is complex and time-consuming. It is better to use numerical methods or programming to find the critical point and evaluate the maximum value of f(x).
Alternatively, you can use graphing software or online graphing tools to plot the function f(x) = x * (64 - x)^3 and visually observe its maximum value on the graph.