Recent analysis has shown that there is a trend in how many people work for a co
ID: 2873843 • Letter: R
Question
Recent analysis has shown that there is a trend in how many people work for a company and what that company’s productivity will be. Suppose you own a company that makes and sells gift baskets for all occasions. Lately, due to the launch of your new website, sales have soared, and you can barely keep up with assembling and shipping the gift baskets all by yourself. You decide to hire an analyst to determine how many employees you should hire to help assemble and ship gift baskets. After a week-long study of your company, the analyst hands you the function
x(n) = 1.42n2 + 213.3n
written on a slip of paper and claims that it predicts the monthly productivity of your company x (number of gift baskets assembled and shipped each month) as a function of the number of employees working for you n. Before the analyst can explain the details of his work, he notices a tow truck taking away his car in the parking lot. Panicking, he rushes out the door, leaving you with the slip of paper in your hand. After several hours of waiting, he doesn’t come back. Luckily, you took Math 144 and know exactly how to interpret this model to maximize your monthly productivity.
(a) How many employees should you hire in order to achieve this goal?
(b) With this number of employees, how many baskets will your company be able to make each month?
(c) Compute x(90) and x(90) (with units) and interpret their significance.
Explanation / Answer
Given: x(n) = -1.42n2 + 213.3n
Hence, x'(n) = (2)(-1.42)n + 213.3 = -2.84n + 213.3
a) Now, at maxima, x'(n) = 0
therefore, -2.84n + 213.3 = 0
=> n = 213.3/2.84 = 75.01 75
We will round it to nearest integer as number of workers will be a whole number.
Hence, to maximize themonthly productivity, 75 employees must be hired.
b) at n = 75,
x(75) = -1.42 (75)2 + 213.3(75)
x(75) = 8010
Hence, 8010 gift baskets can be made and shipped by the team of 75 employees.
c) x(90) = -1.42 (9)2 + 213.3(90)
x(90) = 7695 baskets
x'(90) = -2.84(90) +213.3 = -42.3 basket /employee
x(90) & x'(90) tell us that after the optimum point, the number of baskets that will be made will decrease with an increase in number of employees and that the productivity will decline too.
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