MAT 142 Assignment 4 on Advertising and Sales | Montgomery County Community College

This assignment deals with the attendance rate of a film with a small advertising budget. The attendance rate is given as a function of time and the rate of change in attendance is calculated using calculus. The graph of the attendance rate is also plotted. The attendance rate increases initially before decelerating. The rate of change is positive at t=5, zero at t=10 and negative at t=20.

MONTGOMERY COUNTY COMMUNITY COLLEGE
MAT 142
Assignment 4
Name _____________________________
Every assignment must be typed and a sufficient amount of work shown. Graphs must
be done electronically (I do not want hand drawn graphs.) Directions for graphing using
excel are available in blackboard.
You may ask for help with the assignments.
Advertising and sales. An excellent film with a very small advertising budget must
depend largely on word-of-mouth advertising. If attendance at the film after t weeks is
given by
,
And A is in hundreds of thousands.
a. Use the quotient rule to find and simplify the numerator.
Answer
A' ( t ) = d
dt [ 400 t
t2+ 40 t+100 ]
¿ 400. d
dt [ t
t2 +40 t +100 ]
¿ 400
d
dt [ t ] . ( t2+ 40 t+ 100 ) −t . d
dt [ t2+40 t+100 ]
( t 2+40 t+100 ) 2
¿ 400
t2 +40t +100 − 400 t (2t + 40)
( t2 +40 t+100 ) 2
A' ( t ) =−400(t2−100)
( t2 +40 t+ 100 ) 2
b. Find the rate of change in attendance for t = 5? Write a sentence
interpreting your answer. Use correct units.
Answer
mat\wh\m142\assign\assign4

A' ( 5 )= −400 ( 52−100 )
(52 +40 ( 5 )+100 )2 = −400∗−75
( 25+200+ 100 )2 = 30000
105625 =0.284
The rate of change in attendance for t = 5 is positive implying an
acceleration at this point.
c. Find the rate of change in attendance for t = 10? Write a sentence
interpreting your answer. Use correct units.
Answer
A' ( 10 ) = −400 ( 102−100 )
( 102 +40 ( 10 ) +100 )
2 = −400∗0
( 100+ 4 00+100 )2 = −0
360000 =0
The rate of change in attendance for t = 10 is zero implying there is no
change at this point.
d. Find the rate of change in attendance for t = 20? Write a sentence
interpreting your answer. Use correct units.
Answer
A' ( 20 )= −400 ( 2 02−100 )
(2 02+ 40 ( 2 0 ) +100 )2 = −400∗300
( 400+8 00+100 )2 =−120000
1690000 =−0.071006
The rate of change in attendance for t = 20 is zero implying a deceleration
at this point.
e. Graph A(t).
Answer
mat\wh\m142\assign\assign4

f. Write a paragraph describing how the attendance changes, make sure
you use the results from parts a, b, and c. Use calculus vocabulary.
Answer
Attendance rate increases between time t=0 and time t = 10 before it
starts decelerating. This is evident as can be seen when t = 5 the rate is
positive but it comes to negative at t = 10.
mat\wh\m142\assign\assign4