Question

Separation of Variables:

The rate at which fuel is being used by a Wartsila-Sulzer RTA96-C Turbocharged Diesel Engine is proportional to the amount of fuel already consumed at time t. If there are 2 gallons consumed when t 0 and 3 gallons consumed when t 5, how many gallons will be used when t = 9?

Answer 1

Step 1)

Let's assume that x(t) is the amount of fuel consumed at time t and as given rate at which fuels is being used is proportional to the amount of fuel already consumed at time t hence we can write,

where k is proportionality constant

using separation of variable we can write,

integrate on both the side we can write,

amount of fuel used is never negative hence |x| = x and we can write,

Hence,

we know that e^ln(x) = x hence we can write,

Let's assume that e^c1 = C hence,

we can write,

-----------------------------------------------------1)

Step 2)

As given 2 gallons consumed when t = 0 hence we have x(0) = 2 hence put t = 0 and x(0) = 2 in equation 1) we can write,

Put C = 2 in equation 1) we can write,

-----------------------------------------------------------------2)

As given 3 gallons consumed when t = 5 hence we have x(5) = 3 hence put t = 5 and x(5) = 3 in equation 2) we can write,

Hence,

we know that ln(a^b) = bln(a) hence,

we know that bln(a) = ln(a^b) hence,

Put value of k in equation 2) we can write,

we know that bln(a) = ln(a^b) hence,

we know that e^ln(a) = a hence,

Hence we can say that amount of fuel consumed at time t is given by,

Step 3)

we have,

we have to find the amount of fuels consumed when t = 9 hence put t = 9 we can write,

Hence we can say that amount of fuel consumed at t = 9 is 6.75 gallons