The cost of producing x units of balloons is given by C(x) =140 + 0.125x + 0.0005x^2. If the revenue function is determined by R(x) = 3x, maximum profit is:

it occurs where d/dx(R-C) = 0

I get a maximum profit of 3992.81 at x=2875

profit=Revenue-cost

R=3x
d/dx = 3

C=140+0.125x+0.0005x^2
d/dx=0.001X+0.125

profit=Revenue-cost
p= 3-( 0.001X + 0.125)
p=3 - 0.001x - 0.125
3 - 0.001x - 0.125=0
x=2875

p(x)=R(x)-C(x)
P(2875)= (3x) - (140+0.125x+0.0005x^2)
P(2875)= 3x - 140 - 0.125(2875) - 0.0005(2875)^2
P(2875)=3992.81

To find the maximum profit, we need to determine the optimal production level by maximizing the difference between revenue and cost. We can do this by finding the derivative of the profit function and setting it equal to zero.

The profit function is given by P(x) = R(x) - C(x).

Substituting the given revenue and cost functions:

P(x) = 3x - (140 + 0.125x + 0.0005x^2).

To find the maximum profit, we need to find when the derivative of P(x) is equal to zero.

P'(x) = 3 - 0.125 - 0.001x.

Setting P'(x) equal to zero:

3 - 0.125 - 0.001x = 0.

Simplifying the equation:

-0.001x = -2.875.

Dividing both sides by -0.001:

x = 2875.

The optimal production level for maximum profit is 2875 units.

To find the maximum profit, substitute x = 2875 into the profit function:

P(2875) = R(2875) - C(2875).

P(2875) = 3(2875) - (140 + 0.125(2875) + 0.0005(2875^2)).

P(2875) = 8625 - (140 + 359.375 + 1141.25).

P(2875) = 8625 - 1640.625.

P(2875) = 6984.375.

Therefore, the maximum profit is $6984.375.

To find the maximum profit, we need to calculate the profit function and then determine its maximum value.

The profit function is given by the difference between the revenue function and the cost function:

P(x) = R(x) - C(x) = 3x - (140 + 0.125x + 0.0005x^2)

To find the maximum profit, we need to find the critical points of the profit function. We can do this by taking the derivative of the profit function and setting it equal to zero:

P'(x) = 3 - 0.125 - 0.001x = 0

Simplifying the equation:

-0.001x = -2.875

Dividing both sides by -0.001:

x = 2875

Now we need to check if this critical point is a maximum or a minimum by taking the second derivative of the profit function:

P''(x) = -0.001

Since the second derivative is negative, this means the critical point is a maximum.

Therefore, the maximum profit can be obtained by producing 2875 units of balloons. To find the maximum profit, substitute this value back into the profit function:

P(2875) = 3(2875) - (140 + 0.125(2875) + 0.0005(2875^2))

P(2875) = 8625 - (140 + 359.375 + 1171.09375)

P(2875) = 8625 - 1670.46875

P(2875) = $6954.53

Therefore, the maximum profit is $6954.53.