Cupcakes cost $1.50 each to produce with fixed cost of $50. a) Express the cost C, of producing the cupcakes as a linear function of x cupcakes produced.
b)Find the average cost, C=C/x and simplify appropriately. c) What is the horizontal asymptote of C? Interpret. d) Graph C for x = 5, 10, 15, 25.
a) To express the cost C of producing the cupcakes as a linear function of x cupcakes produced, we need to consider the fixed cost and the variable cost per cupcake.
The fixed cost is given as $50, which means regardless of the number of cupcakes produced, this cost remains constant.
The variable cost per cupcake is stated as $1.50 each. This means that for each cupcake produced, an additional cost of $1.50 is incurred.
To express the total cost C as a linear function of x cupcakes produced, we can use the following formula:
C = mx + b
where m represents the variable cost per cupcake and b represents the fixed cost.
In this case, the equation becomes:
C = 1.50x + 50
b) To find the average cost C = C/x, we substitute the expression for C from part a) into the equation:
C = 1.50x + 50
Dividing this equation by x:
C/x = (1.50x + 50)/x
Simplifying further, we get:
C/x = 1.50 + (50/x)
c) To determine the horizontal asymptote of C, we look at the behavior of the function as x approaches infinity.
In the equation C = 1.50x + 50, the term 1.50x dominates as x increases. This term grows indefinitely with x. Therefore, as x approaches infinity, the cost C also approaches infinity. Hence, there is no horizontal asymptote for C.
Interpretation: The absence of a horizontal asymptote means that the cost of producing cupcakes will continue to increase without bound as more cupcakes are produced.
d) To graph C for different values of x, substitute the given values of x into the equation C = 1.50x + 50 and calculate the corresponding values of C.
For x = 5:
C = 1.50(5) + 50 = 7.50 + 50 = 57.50
For x = 10:
C = 1.50(10) + 50 = 15 + 50 = 65
For x = 15:
C = 1.50(15) + 50 = 22.50 + 50 = 72.50
For x = 25:
C = 1.50(25) + 50 = 37.50 + 50 = 87.50
Now, plot the points (5, 57.50), (10, 65), (15, 72.50), and (25, 87.50) on a graph and connect them.