Texts: The company cannot manufacture more than 3000 pickleball rackets weekly without major expansion to their current operations.
Data points:
1st 2nd 3rd 4th 5th 6th
pickleball rackets: 9.4 12 12.8 14.5 16 x (in hundreds)
20
total cost:
13.4 18 19.8 20.3 23.2 30.4 C(x) (in thousands)
a) Graph the data on the given graph on page 4. Label the axes appropriately with both name and units. Use tick-marks to determine a good scaling (so that the data is spread out) and number the tick-marks. Plot the points with dots and draw a dotted line through the first and fifth data point extending through the whole domain. (The dotted line will not necessarily touch all the data points).
b) What is a reasonable domain considering the context? What is the corresponding range? Give these answers in interval notation. (Hint: Consider what values the number of rackets and total cost can take in the real world. Look at the graph.)
c) Assume the cost model is linear and that the number of rackets, x, is the independent variable. Use the first and fifth data points to write a mathematical model (function) for the weekly cost of manufacturing this racket. Express your model in function notation C(x). For full credit, you need to show how you arrive at your model.
d) Using the cost model from part c, what will the total cost be if they manufacture 18 hundred pickleball rackets weekly? Your cost needs to be accurate to the nearest dollar. Label your answer.
e) Using the cost model from part c, if they want to spend $25 thousand for the manufacture of pickleball rackets weekly, how many of those rackets can they expect to manufacture? Label your answer.
f) Suppose the company sells the pickleball rackets at a price of $3.2 thousand dollars per hundred rackets. If x represents hundred rackets, write the weekly revenue function, R(x). (Recall: Revenue is the price times the number of rackets sold).
g) Write out the weekly profit function, P(x), using the cost model created in part c and the revenue model from part f.
h) What is the weekly profit if the company manufactures and sells 18 hundred rackets weekly? Express your answer in a sentence describing the values in the context of the problem.
