MATH 1211 Written Assignment Unit 4- University of the People
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Course
MATH 1211
Institution
MATH 1211
MATH 1211 Written Assignment Unit 4- University of the PeopleMATH 1211 Written Assignment Unit 4- University of the PeopleMATH 1211 Written Assignment Unit 4- University of the People
1. Chains Inc. is in the business of making and selling chains. Let be the
number of miles of chain produced after hours of production. Let be
the profit as a function of the number of miles of chain produced and
let be the profit as a function of the number of hours of production.
Suppose the company can produce 3 miles of chain per hour and suppose
their profit on the chains is $4000 per mile of chain. Find and interpret
(use complete sentences) each of the following (include units), , ,
and . How does relates to and ?
solution
It is given that c(t) represents the number of miles of chain produced after t hours
of production.
The company can produce 3 miles of chain per hour, thus, c(t)=3t.
Differentiate the function c(t)=3t with respect to t using the power rule of the
derivative to obtain the value of c’(t).
d d
c ( t )= ( 3 t )
dt dt
d
¿3 (t )
dt
c’(t) =3
it is known that the derivative of any function refers to the rate of change of the
function with respect to the respective variable.
Hence, c’(t)=3 shows that the number of miles in every increase in t is 3 miles per
hour.
It is given p(c) represents the profit as a function of the number of miles of chain
produced.
The company’s profit on the chains is $4000 per mile of chain, thus, p(c) will the
product of 4000 and the function for the number of miles c(t).
P(c) = 4000 . c(t)
The obtained function can be written as p(c)=4000c.
, Differentiate the function p(c)=4000c with respect to c by using the power rule of
the derivative to obtain the value of p’(c).
d d
p ( c ) = ( 4000 c )
dc dc
d
P’(c) =4000 dc (c )
=4000
Hence, p’(c)=4000 profit/mile shows that the profit increases by 4000 per mile of
chain produced.
It is given that q(t) represents the profit as a function of the number of hours of
production.
As per the given statement for q(t), the function will be obtained by multiplying
4000 and the function c(t).
profit miles
q ( t )=4000 x c (t )
miles hours
profit
¿ ( 4000 x 3 t )
hours
profit
¿ 12000 t
hours
Differentiate the function q(t)=12000t with respect to t by using the power rule of
the derivate to obtain the value of q’(t).
q’(t) = 12000
hence, q’(t)=12000 shows that the profit increases by 12000 per hour.
It is obtained that c’(t)=3, p’(c)=4000, and q’(t)=12000. After multiply 3 mile/hour
by 4000 profit/mile the product will be 12000 profit/hour.
Therefore, the product of the functions c’(t) and p’(c) is equal to the function q’(t).
Hence, the relation will be,
q’(t) = c’(t) . p’(c)
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