Answer:
a) C(x) = 1.33 + 0.097x
b) Fixed Initial cost = $1.33
c) C(1200) = $ 117.73
Step-by-step explanation:
a) Let's first define our x variable and y variable as:
x: Number of cups of coffee produced
y: Cost of producing
y is a function of x that in this problem is called C(x) so y = C(x).
No we are told that C(x) is a linear function. All linear functions follow the rule:
C(x) = mx+b
where m is the slope of the line and b is the intercept in the y - axis or the value of the function when x=0 . To find a formula for C(x) we can use the information given because these are two points of the line where
Point 1
x1= 300 and y1 = 30.43
Point 2
x2= 500 and y2 = 49.83
With these two points we can find the slope with the formula
m= y2-y1/x2-x1 = (49.83-30.43)/(500-300) = 19.42/200 = 0.097
so we have that;
C(x) = mx+b = 0.097x+b.
Now we have to know b the intercept in y.For this problem this is equivalent to the cost that we would have to pay if we did not produced any cup so b is our fixed initial cost. Because we have a point, we can replace it in the equation and solve for b. It doesnt matter which point we use.
C(x) = 0.097x + b
b = C(x) - 0.097x
With Point 2 = x = 500 and C(x) = 49.83
b = C(x) - 0.097x
b = 49.83 - (0.097 * 500) = 49.83 -48.5 = 1.33
So the final formula for C(x) is
C(x) = 0.097x + 1.33
b) As I said before, the initial cost or fixed cost is the cost incurred if we would not produce anything or mathematically when x = 0
C(x) = 0.097x + 1.33
C(0) = 0.097*0 + 1.33 = 0+1.33 = 1.33
The fixed cost is $ 1.33 that is the same as b parameter.
c) Now that we have an equation for C(x) we only need to replace for the point x = 1200
C(x) = 0.097x + 1.33
C(1200) = (0.097*1200) + 1.33 = 116.4 +1.33 = $ 117.73
