The function Pt gives gas prices in units of dollars per gal

The function P(t) gives gas prices (in units of dollars per gallon) as a function of t the year (in A.D. or C.E.), and g(t) is the gas consumption rate measured in gallons per year by a driver as a function of their age. The function g is certainly dierent for dierent people. Assuming a lifetime is 100 years, what function gives the total amount spent on gas during the lifetime of an individual born in an arbitrary year t? Is the operator that maps g to this function linear?

Solution

P(t) gives the gas prices (in units of dollars per gallon) as a function of t.

g(t) gives the gas consumption rate (gallons per year)

Therefore, total amount spent on gas by a person in an year would be given by the product of P(t) and g(t)

Hence, total amount spent on gas (per year) = P(t)g(t)

Since, we have been given lifetime of a person as 100 years (which can be considered average lifetime), therefore, total amount spent on gas by an individual per year would be obtained by multiplying P(t)g(t) by 100.

Hence the total amount spent on gas during the lifetime of an individual born in an arbitrary year t is 100P(t)g(t).

The operator that maps, g(t) into 100P(t)g(t) is 100P(t).

We do not know if P(t) is a linear function. Therefore, we cannot say if the operator 100P(t) would be linear. It will depend on P(t). If P(t) is linear then 100P(t) will also be linear, otherwise it wonn\'t be linear.