math question and I need urgent help

Question

The value M (in dollars) of a motorcycle t years after it was purchased new can be estimated using the function:


M(t)=3500/t+500

 

a. Estimate the motorcycle's value 5 years after it was purchased. $ ___

b. What does the value of the motorcycle approach as time passes? $ ___

Answer 1

Part A is too easy, so I will move on to the next part.

 

 

Part b)

 

As time passes, the value of t increases infinity.  As this happens, the value of M(t) get very close to a certain constant.  This constant is the horizontal asymptote.

 

You can find this constant number in two ways. 

 

The first way is to evaluate M(t) at various t values.  t=0 , t=100 , t=1000 , t=100000, and t=100000000000.

 

Note: t=0 is the time of purchase

 

Notice how the value of t gets larger and larger as it is evaluated?  Eventually, the M(t) values gets closer to a constant number during this increase.

 

The second method is to find the horizontal asymptote.  What do you notice about the degree of the numerator and denominator in the rational function?  The degree of the numerator is less than the degree of the denominator.  Because of this, the horizontal asymptote is

 

y = 0

 

Therefore, as time passes, the value of the motorcycle approaches zero.  In other words, the motorcycle will be worthless over time.  This seems logical in the real world because products eventually become obsolete as new products are introduced.