Question

The manager of a restaurant found that the cost to produce 300 cups of coffee is ​$30.43​, while the cost to produce 500 cups is ​$49.83. Assume the cost​ C(x) is a linear function of​ x, the number of cups produced.
a) Find the formula for C(x)
b) What is the fixed (initial cost)
c) Find the total cost of producing 1200 cups

Answer 1

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