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
So the objective = f(x)=xy and the constraint = x+5y=5000
Now number 3 says rewrite the objective function in terms of the variable x only. Would it be y=x?
4) For which value of x does the objective function attain its maximum? Do I have to take the derivative?
5) What is the corresponding value of y
6) What is the vale of the maximum product of the two numbers?
3)
from x+5y=5000, we get y=(5000-x)/5
substitute into f(x) to get:
f(x)=xy=x(5000-x)/5
4)
Take derivative and equate to zero. I get x=2500,
5)
find y using equation from 3)
6)
x*y
Thank You
did you get 1000-2x/5 as a derivative for x(5000-x)/5?
Yes, your derivative is correct.
The financial officer for an accounting firm allows Dh 50,000 for computer supplies in the annual budget. After 6 months, Dh 26,200 has been spent on supplies. Is this figure within 55% of the annual budget?
3) No, if you rewrite the objective function in terms of x only, it would be f(x) = x(5000 - 5x).
4) Yes, to find the maximum value of the objective function, you would need to take the derivative of f(x) and set it equal to zero to solve for x.
5) Once you find the value of x that maximizes the objective function, substitute it back into the constraint equation x + 5y = 5000 to solve for y.
6) After finding the values of x and y, you can compute the maximum product by multiplying x and y together.
1) The objective equation is the equation that represents the quantity you want to maximize or minimize. In this case, the objective equation is f(x) = xy, representing the product of the two numbers x and y.
2) The constraint equation is the equation that represents the condition or restriction that needs to be satisfied. In this case, the constraint equation is x + 5y = 5000, representing the condition that the sum of x and five times y is equal to 5000.
3) To rewrite the objective function in terms of the variable x only, you can substitute the constraint equation into the objective equation. Since x + 5y = 5000, you can solve this equation for y and substitute it into the objective equation: f(x) = x(5000 - x)/5.
4) To find the value of x that maximizes the objective function, you can take the derivative of the objective function with respect to x and set it equal to zero. In this case, you can differentiate f(x) = x(5000 - x)/5, and solving for x, you will find the value that maximizes the function.
5) Once you find the value of x that maximizes the objective function, you can substitute it back into the constraint equation x + 5y = 5000 to find the corresponding value of y.
6) Finally, you can substitute the values of x and y into the objective equation f(x) = xy to calculate the maximum product of the two numbers.