Compute the maximum product for two positive numbers x and y with the property that the sum of the first plus five times the second is 5000.
1) Indicate the objective equation
2) Indicate the constraint equation
I put x+5y=5000. Is this correct.
objective function f(x)=xy
constraint: x+5y=5000
For the same question:
3) Rewrite the objective function in terms of the variable x only.
I out x=5000-5y
4) For which value of x does the objective function attain its maximum?
Do I have to take the derivative?
Yes, you correctly identified the constraint equation as x + 5y = 5000. This equation represents the property that the sum of the first number (x) plus five times the second number (5y) is equal to 5000.
Now, let's move on to identifying the objective equation, which will allow us to compute the maximum product of x and y. To do this, we need to maximize the product, which means maximizing the value of xy.
To find the maximum value of the product, we can use the method of substitution. Rearrange the constraint equation to solve for x:
x = 5000 - 5y
Now substitute this expression for x into the objective equation, which is xy:
xy = (5000 - 5y)y
Simplifying this expression further will give us the objective equation.