Profit Calculation for Kayak Manufacturing Company
a) What is the rate of increase in revenue when production output is increasing at 3 kayaks per day from 40 kayaks per day?
Show all the steps to calculate the rate of increase in revenue.
b) How much does it cost to make each kayak and what is the profit function for the company?
a) The rate of increase in revenue is approximately $2228.67 per day.
To find the rate of increase in revenue, we need to differentiate the revenue function with respect to production output. The revenue function is given by R = 750x - x^2/30, where x represents the production output in one day. Taking the derivative of the revenue function with respect to x, we get: dR/dx = 750 - 2x/30.
To find the rate of increase in revenue, we substitute the given production output into the derivative. Since production is increasing at 3 kayaks per day when it is at 40 kayaks per day, we substitute x = 40 and dx/dt = 3 into the derivative: dR/dt = (750 - 2(40)/30) * 3. Simplifying the expression, we get dR/dt ≈ 2228.67. Therefore, the rate of increase in revenue is approximately $2228.67 per day.
b) The cost to make each kayak is $124 and the profit function is P = 626x - x^2/30.
To calculate the cost of making each kayak, we are given that each kayak costs $124. The revenue function is R = 750x - x^2/30, where x represents the production output in one day. To find the profit, we subtract the cost from the revenue: Profit = Revenue - Cost. Simplifying the expression, we get Profit = 626x - x^2/30. Therefore, the profit function for the company is P = 626x - x^2/30.
In summary, the rate of increase in revenue for the kayak manufacturing company is approximately $2228.67 per day when the production output is increasing at 3 kayaks per day from 40 kayaks per day. This calculation is based on the revenue function R = 750x - x^2/30, where x represents the production output in one day.
On the other hand, the cost to make each kayak is $124, and the profit function for the company is P = 626x - x^2/30. By subtracting the cost from the revenue, the company can determine its profit margin and make informed business decisions moving forward.