A company manufactirung snowboards has fixed costs of $200 per day and total cost of $3800 per day at a daily output of 20 boards.
(A( Assuming that the total cost per day, C(x), is linearly related to the taotal out per dau,x, write an equation for the xost function
(B) The cost per board for an output of x boards is given by C(x)=C(x)/x, find the average cost function.
(C) What does the average cost per board tend to as production increases?
(D) sketch a graph of the average cost function, including any asympototes, for 1,=x,=30
A. $3800-200 = $3600 = Variable cost.
$3600/20Boards = $180/board.
C(x) = 180x + 200.
C(x) = 180x + 200.
(A) To find the equation for the cost function, we need to determine the relationship between the total cost and the total output per day.
We are given that the fixed costs are $200 per day, which means they do not change with the production level. Therefore, the variable costs must be $3800 - $200 = $3600.
The total cost per day, C(x), can be given as:
C(x) = fixed costs (FC) + variable costs per unit (VC) * total output (x)
Since the fixed costs are $200 per day and the variable costs per unit are $3600, we have:
C(x) = $200 + $3600 * x
(B) The cost per board for an output of x boards is given by C(x) / x.
So, the average cost function is:
AC(x) = C(x) / x
AC(x) = ($200 + $3600 * x) / x
AC(x) = $200/x + $3600
(C) To find what the average cost per board tends to as production increases, we take the limit as x approaches infinity for the average cost function:
lim(x->∞) AC(x) = lim(x->∞) ($200/x + $3600)
As x approaches infinity, the term $200/x approaches 0, so we are left with:
lim(x->∞) AC(x) = $3600
Therefore, as production increases, the average cost per board tends to $3600.
(D) To sketch a graph of the average cost function, we can plot points for various values of x and connect them.
For x = 1, the average cost per board is:
AC(1) = $200/1 + $3600 = $3800
For x = 30, the average cost per board is:
AC(30) = $200/30 + $3600 = $2066.67
We can plot these points and sketch the graph accordingly. Note that the average cost function will have a horizontal asymptote at $3600.
Please note that the graph cannot be directly generated by text, but you can plot the points and connect them on a graphing software or draw it manually on paper.
(A) To write the cost function, let's start by finding the slope (rate of change) of the total cost per day with respect to the total output per day. In this case, the fixed cost remains the same and only the variable cost changes. So, we can calculate the variable cost per day as follows:
Variable Cost = Total Cost - Fixed Cost
Variable Cost = $3800 - $200
Variable Cost = $3600
Now, we can find the variable cost per board:
Variable Cost per Board = Variable Cost / Total Output
Variable Cost per Board = $3600 / 20
Variable Cost per Board = $180
Using the slope-intercept form of a linear equation (y = mx + b), where y is the total cost per day (C(x)), m is the slope (variable cost per board), and x is the total output per day, we can write the cost function as:
C(x) = mx + b
C(x) = $180x + $200
Therefore, the equation for the cost function is C(x) = $180x + $200.
(B) To find the average cost function, we divide the total cost (C(x)) by the total output per day (x):
Average Cost Function = C(x) / x
Average Cost Function = ($180x + $200) / x
(C) As production increases, the average cost per board tends to decrease. This is because fixed costs are spread over a larger number of boards, resulting in a lower cost allocated to each board.
(D) To sketch the graph of the average cost function, we need to plot points for different values of x (total output per day) and calculate the corresponding average cost per board using the average cost function:
For x = 1,
Average Cost = ($180(1) + $200) / 1
Average Cost = $380
For x = 30,
Average Cost = ($180(30) + $200) / 30
Average Cost = $206
Plotting these points on a graph and connecting them with a smooth curve will give you the graph of the average cost function. Additionally, as x approaches infinity, the average cost per board will approach the variable cost per board, which is $180 in this case. Hence, there will be a horizontal asymptote at y = $180.