A company manufacturing surfboards has fixed costs of $300 per day and
total costs of $5100 per day for a daily output of 20 boards. Assume the total cost per day C(x) is linearly related to the total output per day x. Write an equation for the cost
function, and write an equation for the average cost function C(x)=C(x)/x. What does the average cost per board tend to as production increases (assume production
output goes to infinity)?
To find the equation for the total cost function, we need to consider the fixed costs and the variable costs. In this case, we know that the fixed costs are $300 per day and the total costs are $5100 per day for a daily output of 20 boards.
First, let's find the variable costs by subtracting the fixed costs from the total costs:
Variable costs = Total costs - Fixed costs
Variable costs = $5100 - $300
Variable costs = $4800
Now we can find the cost per board by dividing the total costs by the daily output:
Cost per board = Total costs / Daily output
Cost per board = $5100 / 20
Cost per board = $255
Thus, the cost function can be expressed as:
C(x) = 300 + 255x
To find the average cost function, we need to divide the cost function by the output:
Average cost function = C(x) / x
Average cost function = (300 + 255x) / x
As the production output increases (as x approaches infinity), we can observe the behavior of the average cost per board. Dividing a constant (300) by an increasing value (x) will cause the average cost per board to approach zero. Therefore, the average cost per board tends to zero as production increases.