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

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

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