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Example: The monthly demand, D , for a product, is. where p is the price per unit of the product. The price per unit, p , for the product is p = 2000 – 10 t , where t is the number of months past January 1995. Write the monthly demand, D , as a function of t. D p.

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Example: The monthly demand, D, for a product, is

where p is the price per unit of the product.

The price per unit, p, for the product is p = 2000 – 10t, where t is the number of months past January 1995.

Write the monthly demand, D, as a function of t.

D

p

Note, D is a function of p,

D(p)

and p is a function of t.

p(t)

t

(D p)(t) =

Composite Functions: Application

Compute (D p)(t) = D(p(t)).


(D p)(t) =

This is now a function of demand with respect to t, so can

be relabeled,

D(t) =

6250 =

Composite Functions: Application

When will the monthly demand reach 6,250 units?

6250(2000 – 10t) = 5000000,

12500000 – 62500t = 5000000,

- 62500t = - 7500000,

t = 120 months

The monthly demand will reach 6,250 units in January 2005.

Slide 2


Try: 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).

Jot down the figures above and click to see the questions!

d

x

300 feet

Composite Functions: Application

Slide 3


d launching point of a balloon. The balloon is rising is rising at a rate of 10 feet per second.

x

300 feet

Composite Functions: Application

(a) Express d as a function of x. Hint: Use the Pythagorean Theorem.

(b) Express x as a function of t.

x(t) = 10t

(c) Express d as a function of t.

(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.

Slide 4


Composite Functions: Application launching point of a balloon. The balloon is rising is rising at a rate of 10 feet per second.

END OF PRESENTATION

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