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Somewhat complex Composite function problems - Help please!

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a) b) Dec 30, 2017

#1
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a)  For the last pair, here's an example and explanation.

Imagine someone is putting air in a spherical balloon. The radius of the balloon is increasing as time passes. Let's say we have a function that can tell us the radius at any point in time....

the radius of the balloon  =  g(x)   , where  x  is the time in some units, let's say seconds.

So if someone asked, "What is the radius of the balloon when the time is  7  seconds?"

It would just be  g(7) .

We also have a function that tells us the volume, if we know what the radius is....

the volume of the balloon  =  h(x)  , where  x  is its radius in inches.

So if someone asked, "What is the volume of the balloon when the radius is  3 inches?"

It would just be  h(3) .

Now what if someone asked, "What is the volume of the balloon when the time is  7  seconds?"

We know the radius at any given time:  When the time is  7 , the radius is  g(7) .

We know the volume at any given radius:  When the radius is  g(7) , the volume is  h( g(7) ).

So when the time is  7  seconds, the volume is  h( g(7) )

So....

h( g(x) )  =  the volume as a function of time.

Dec 30, 2017
#2
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6)

Let  k(t)  =  distance between lighthouse and the ship....where t is in hours

Let d(t)   = distance the ship had traveled since 11AM  =  35t  [this is the answer to (b) ]

Expressing  k  as a function  of  d  we have that

k  =  sqrt  [ 10^2  + (d(t))^2  ]      [ this is the answer to (a)  ]

(c)  ( k °  d)  =  k (d(t))  =    sqrt  [ 10^2  + (35t)^2  ]

k actually represents the hypotenuse length of a right triangle where  10 km  is one leg and 35t km the other leg   Dec 31, 2017
edited by CPhill  Dec 31, 2017