Mary's utility function is U(b,c)= b+ 100c - c^2. b takes up 1 sq ft and c takes up 4 sq ft.
a) if Mary has only 144 sq ft, how much of c would she grow?
b) if mary grows both b and c, how large must her garden be in sq ft?
a) To determine the amount of c that Mary would grow, we need to find the optimal value of c that maximizes her utility function U(b, c) given the constraint of having only 144 sq ft.
Let's set up the optimization problem:
Maximize U(b, c) = b + 100c - c^2
subject to the constraint b + 4c = 144
To solve this problem, we can use the method of Lagrange multipliers. Let λ be the Lagrange multiplier.
Set up the Lagrangian:
L(b, c, λ) = U(b, c) + λ(b + 4c - 144)
Take partial derivatives with respect to b, c, and λ:
∂L/∂b = 1 + λ = 0
∂L/∂c = 100 - 2c + 4λ = 0
∂L/∂λ = b + 4c - 144 = 0
Solve these equations simultaneously:
1 + λ = 0 --> λ = -1
100 - 2c + 4λ = 0 --> -2c - 4 = 0 --> c = -2
b + 4c - 144 = 0 --> b - 8 - 144 = 0 --> b = 152
As c cannot be negative in this context, we discard the negative solution. Therefore, Mary would not grow any c plants given her limited garden size of 144 sq ft.
b) To determine how large Mary's garden must be in square feet if she wants to grow both b and c, we need to find the combined area of b and c.
b takes up 1 sq ft, and c takes up 4 sq ft. Therefore, the total area required for growing both b and c is 1 sq ft for b and 4 sq ft for c, which sums up to 5 sq ft.
Therefore, Mary's garden must be at least 5 sq ft in order to accommodate growing both b and c.
To answer these questions, we need to understand Mary's utility function and calculate the values of b and c that maximize her utility.
a) To find out how much of c Mary would grow if she has only 144 sq ft, we need to maximize her utility function U(b,c) under the constraint that c must take up 4 sq ft. Mathematically, this is an optimization problem.
1. Define the optimization problem: Maximize U(b,c) = b + 100c - c^2
subject to the constraint: c = 4
In this case, the constraint is that c must take up 4 sq ft.
2. Substitute the constraint into the utility function: U(b) = b + 100(4) - 4^2 = b + 400 - 16 = b + 384
3. Now, we only have one variable (b) in the utility function. We can maximize U(b) by taking the derivative with respect to b, setting it equal to zero, and solving for b.
dU(b)/db = 1 + 0 = 1
Setting dU(b)/db = 0, we get 1 = 0, which is not possible. This means that the derivative is always positive and the utility function is increasing. Therefore, there is no maximum value for U(b) under the given constraint.
Hence, there is no specific answer to a) since the utility function does not have a maximum value within the given constraint. Mary could grow any amount of c between 0 and 4 sq ft.
b) To calculate the total garden size required when Mary grows both b and c, we need to find the values of b and c that maximize her utility function without any constraint on the size of b or c.
1. Define the optimization problem: Maximize U(b, c) = b + 100c - c^2
2. We need to maximize U(b, c) by taking the partial derivatives of U(b, c) with respect to both b and c, setting them equal to zero, and solving for b and c.
Partial derivative with respect to b:
dU(b, c)/db = 1 + 0 = 1
Partial derivative with respect to c:
dU(b, c)/dc = 100 - 2c
Setting both derivatives equal to zero:
1 = 0 (not possible)
100 - 2c = 0
2c = 100
c = 50
Plugging c = 50 back into the utility function:
U(b, 50) = b + 100(50) - (50)^2
= b + 5000 - 2500
= b + 2500
3. In this case, since there is no constraint on the size of b or c, the utility function is maximized when both b and c take up the maximum possible area. Therefore, Mary's garden must be a total of 144 sq ft.
Hence, the answer to b) is that Mary's garden must be 144 sq ft in total to maximize her utility when growing both b and c.