Suppose Fred has a utility function of the form U = X ½ Y ½ . Fred has an income of $100, good X costs $1.00, and good Y costs $2.00. How much of good X and how much or good Y will Fred buy? Be sure to show all your work.

Ive aready drawn my graph and i know that under my budget constraint i could buy 100 of X only and 50 of y only. i also know that my slop is -2. i just can not figure out how to solve to find how much of good X and Y he will buy.
thanks

see my post to your earlier post.

To find out how much of good X and good Y Fred will buy, we need to identify his optimal consumption bundle that maximizes his utility under the given budget constraint.

Here's how we can find the solution step by step:

Step 1: Determine the budget constraint:
Fred's income is $100, and good X costs $1.00 while good Y costs $2.00. Using this information, we can set up the equation for the budget constraint:
X + 2Y = 100

Step 2: Rewrite the utility function:
Fred's utility function is U = X^(1/2) * Y^(1/2). Let's rewrite this in terms of X or Y to simplify:
U^2 = X * Y

Step 3: Solve for Y in terms of X:
Rearrange the budget constraint equation to solve for Y:
Y = (100 - X)/2

Step 4: Substitute the value of Y into the utility function:
Substitute (100 - X)/2 for Y in the utility function U^2 = X * Y:
U^2 = X * ((100 - X)/2)

Step 5: Maximize the utility function:
To find the optimal consumption bundle, we need to maximize the utility function U^2 with respect to X. Differentiate U^2 with respect to X and set it equal to zero to find the critical point:

d(U^2)/dX = (100 - 3X)/2 = 0

Step 6: Solve for X:
Solve the equation (100 - 3X)/2 = 0 for X:
100 - 3X = 0
3X = 100
X = 100/3

Step 7: Find the value of Y:
Substitute the value of X into the budget constraint equation to find Y:
Y = (100 - X)/2
Y = (100 - (100/3))/2
Y = (200/3)/2
Y = 100/3

So, Fred will buy approximately 33.33 units of good X (X ≈ 100/3) and 33.33 units of good Y (Y ≈ 100/3) to maximize his utility given the budget constraint.