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