Section 1.4 Problem 35

Section 1.4 Problem 35

(a) She has 60 hours a week to spend on meetings.
If she spends all of the time on client meetings which take 3 hours each and no co-worker meetings, how many of these client meetings can she have? 20.
If she spends all of the time on co-worker meetings which take 2 hours each and no client meetings, how many of these co-worker meetings can she have? 30.
A graph would pass through the points (0, 20) and (30, 0). Since there is a linear relationship, we have the following. Here x is the number of co-worker meetings held and y is the number of client meetings.

(b) If she holds x co-worker meetings at 2 hours each and also y client meetings at 3 hours each, then the relationship between x and y is 2x + 3y = 60. Writing this in the form of y as a function of x, we have

3y = 60 − 2x

y = 60 − (2/3)x

(c) The intercepts have the following representation:
The x-intercept (30, 0) means she used all of her 60 hours on 30 co-worker meetings and held 0 client meetings.
The y-intercept (0, 20) means she used all of her 60 hours on 20 client meetings and held 0 co-worker meetings.
To understand the slope, think about the average rate of change. It might be helpful to look at a table, where x changes by 3's.

co-worker meetings, x	client meetings, y
0	20
3	18
6	16
9	14
12	12
15	10
18	8
21	6
24	4
27	2
30	0

What does the slope, or average rate of change −2/3 mean, and what are the units of the slope?
The change in x is 3 co-worker meetings and the change in y is −2 client meetings, so the slope has units

     - 2 client meetings
 m = --------------------
     3 co-worker meetings 

It means that for every 3 (two hour) co-worker meetings she adds, she must sacrifice 2 (three 
hour) client meetings.

One could also say that for every 2 (three hour) client meetings she takes on, 
she has to give up 3 (two hour) co-worker meetings since

     - 2 client meetings        2 client meetings
 m = --------------------  = ----------------------
     3 co-worker meetings    - 3 co-worker meetings

(d) If co-worker meetings are shortened to 1.5 hours, the equation changes from 2x + 3y = 60 to 1.5x + 3y = 60. The slope changes from −2/3 to −1.5/3, or, more simply −1/2. In slope-intercept form we have y = 20 −(1/2)x. The y-intercept remains the same.
She can now squeeze in 40 co-worker meetings in a week rather than 30 (since they're shorter), if she chooses to do 0 client meetings.
Now for every 2 (1.5 hour) co-worker meetings she adds, she must sacrifice 1 (three hour) client meeting.

co-worker meetings, x	client meetings, y
0	20
2	19
4	18
6	17
8	16
10	15
12	14
14	13
16	12
18	11
20	10
...	...
34	3
36	2
38	1
40	0

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Last updated: September 26, 2003
URL: http://users.ipfw.edu/lamaster/ma153/section_1.4problem35.htm
Contact: John LaMaster, Instructor, (260) 481-5430
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