Respuesta :
Given:
Marginal cost for making 1 mask = $0.50
Total cost to make 100 masks = $62
Revenue per mask = $3
To find:
(a) Linear cost function C(x)
(b) Revenue function R(x)
(c) Break even point
(d) Interval of profit & loss
Solution:
(a) We know that linear cost function is given by,
Total cost = Fixed cost per unit + (Marginal cost per unit)*(Number of units)
Let 'x' denote the number of units
[tex]\Rightarrow[/tex] C(x) = b + mx
It is given that, m = $0.50
[tex]\Rightarrow[/tex] C(x) = b + 0.5x
It is also given that the total cost of making 100 masks is $62
[tex]\Rightarrow[/tex] C(100) = $62
[tex]\Rightarrow[/tex] b +0.5(100) = 62
[tex]\Rightarrow[/tex] b + 50 =62
[tex]\Rightarrow[/tex] b = 62 - 50
[tex]\Rightarrow[/tex] b = 12
[tex]\Rightarrow[/tex] C(x) = 12 + 0.5x
This is the linear cost function, C(x)
(b) We know that,
Total Revenue = Revenue per unit * Number of units
It is given that a mask is sold for $3, i.e., revenue per mask is $3
Let 'x' denote the number of units
Then the revenue function is given by,
R(x) = 3x
This is the revenue function, R(x)
(c) We know that, break even point refers to the point where total cost is equal to the total revenue. Thus, at break even point, we have,
C(x) = R(x)
[tex]\Rightarrow[/tex] 12 + 0.5x = 3x
[tex]\Rightarrow[/tex] 3x - 0.5x = 12
[tex]\Rightarrow[/tex] 2.5x = 12
[tex]\Rightarrow x=\frac{12}{2.5}[/tex]
[tex]\Rightarrow[/tex] x = 4.8
Since, 'x' denotes the number of masks, it must be a whole number and not a fraction. Thus, we will round off our value to get the break even point as 5 masks.
That is, 5 masks must be made to break even.
(d) We know that the profit function is given as the difference of revenue function and cost function. That is, we have,
P(x) = R(x) - C(x)
[tex]\Rightarrow[/tex] P(x) = 3x - (12 + 0.5x)
[tex]\Rightarrow[/tex] P(x) = 3x - 12 - 0.5x
[tex]\Rightarrow[/tex] P(x) = 2.5x -12
Now, we know that there is profit when the value of the profit function is positive & there is loss when the value of profit function is negative. Thus, we can calculate the intervals of profit and loss by finding the intervals where the profit function is positive & negative respectively. Alternatively, since the break even point denotes the point where the value of the profit function is 0, we can find the intervals of profit and loss as the intervals greater than and lesser than the break even point respectively.
That is, since the break even point is x = 4.8, the interval of profit is given as x > 4.8 & the interval of loss is given as x < 4.8
Taking into account that 'x' denotes the number of masks and thus must be a whole number, we have the intervals of profit and loss as,
Loss:= [tex]x \in [0,4][/tex]
Profit:= [tex]x \in [5, \infty)[/tex]
Final answer:
(a) Linear Cost function: C(x) = 12 + 0.5x
(b) Revenue function: R(x) = 3x
(c) 5 masks must be made to break even
(d) Interval of profit: [tex]x \in [5, \infty)[/tex], Interval of loss: [tex]x \in [0,4][/tex]
