The demand function for a bottle of hand lotion is p(x)=0.78+0.0003x, where x is the number of bottles sold and p is the price, in dollars. The cost function is c(x)=480-0.32x+0.0005x^2. Find the marginal profit when 700 bottles of hand lotion are sold.
To find the marginal profit when 700 bottles of hand lotion are sold, we need to calculate the marginal profit function first.
The profit function can be obtained by subtracting the cost function from the revenue function. The revenue function is given by multiplying the demand function with the price.
Let's first find the revenue function:
Revenue (R) = Price (p) × Quantity (x)
R(x) = p(x) × x = (0.78 + 0.0003x) × x = 0.78x + 0.0003x^2
Now, let's calculate the cost function:
Cost (C) = 480 - 0.32x + 0.0005x^2 = 0.0005x^2 - 0.32x + 480
To find the profit (P), we subtract the cost from the revenue:
P(x) = R(x) - C(x)
P(x) = (0.78x + 0.0003x^2) - (0.0005x^2 - 0.32x + 480)
P(x) = 0.0003x^2 + 0.78x - 0.0005x^2 + 0.32x - 480
Now, to find the marginal profit at x = 700, we need to calculate the derivative of the profit function with respect to x and then evaluate it at x = 700.
P'(x) = d(P(x))/dx = d(0.0003x^2 + 0.78x - 0.0005x^2 + 0.32x - 480)/dx
P'(x) = 0.0006x + 0.78 + 0.32
P'(x) = 0.0006x + 0.32 + 0.78
P'(x) = 0.0006x + 1.1
Now, substitute x = 700 into the derivative equation to find the marginal profit at 700 bottles sold:
P'(700) = 0.0006(700) + 1.1
P'(700) = 0.42 + 1.1
P'(700) = 1.52
Therefore, the marginal profit when 700 bottles of hand lotion are sold is $1.52.
profit = revenue - cost
= demand * price - cost
= x*p(x)-c(x)
= x(.78+.003x) - (480-.32x+.0005x^2)
= 0.0025x^2 + 1.1x - 480
marginal profit = dp/dx = 0.005x^2 + 1.1
p'(700) = 0.005*700^2 + 1.1 = 2451.1