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)
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)
