composite functions: application the price per unit, p, for the product is p = 2000 – 10t, where t...
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Composite Functions: Application
The price per unit, p, for the product is p = 2000 – 10t, where t is the number of months past January 2010.
Example 1:
The monthly demand, D, for a product, is
5,000,000D
p
where p is the price per unit of the product.
Write the monthly demand, D, as a function of t.
Composite Functions: Application
Compute (D p)(t) = D(p(t)).
Note, D is a function of p, D(p)D
pand p is a function of t.
tp(t)
5,000,000D
p 2000 10p t
(2000 10 )D t 5000000
2000 10t
Composite Functions: Application
(D p)(t) =t102000
000,000,5
This is now a function of demand with respect to t, so can be relabeled,
5,000,000( )
2000 10D t
t
Composite Functions: Application
When will the monthly demand reach 6,250 units?
5,000,000( )
2000 10D t
t
5,000,0006250
2000 10t
Composite Functions: Application
6250(2000 – 10t) = 5000000
12500000 – 62500t = 5000000
- 62500t = - 7500000
t = 120 months
The monthly demand will reach 6,250 units in January 2005.
5,000,0006250
2000 10t
Composite Functions: Application
Example 2: An observer on the ground is 300 feet away from the launching point of a balloon. The balloon is rising is rising at a rate of 10 feet per second.
Let d = the distance (in feet) between the balloon and the observer.
Let t = the time elapsed (in seconds) since the balloon was launched.
Let x = the balloon's altitude (in feet).
300 feet
xd
Composite Functions: Application
(a) Express d as a function of x.
Hint: Use the Pythagorean Theorem.
300 feet
xd
22300 xxd
2 2 2300d x
2 2300d x
Composite Functions: Application
300 feet
xd
(b) Express x as a function of t.
x(t) = 10t
The balloon is rising is rising at a rate of 10 feet per second.
x = the balloon's altitude (in feet).
Composite Functions: Application
300 feet
xd
(c) Express d as a function of t.
( )d x t d x t
x(t) = 10t
22300 xxd
2 2300 (10 )t
Composite Functions: Application
300 feet
xd
(d) Use the result found in (c) to determine how long it takes from launching for the balloon to be 500 feet from the observer.
It takes 40 seconds.
2 2( ) 300 (10 )d t t
2 2 2500 300 100t
2 22500 300
100t
40t
Composite Functions: Application