Question

Assume the model is dc/dt =kc(t) where c(t) is the number of cars produced at time t. if the factory's intial production was $100$ cars and in $10$ years the production incraesed to $200$ cars, then the solution of the model gives C(t)=αe kt where α= and k=

Answer 1

To find the constants α and k in the equation C(t) = αe^{kt}, we use the initial car production of 100 cars and the increased production to 200 cars after 10 years. We deduce α to be 100 and calculate k as ln(2)/10.

Step-by-step explanation:

The student is asking to solve a differential equation dc/dt = kc(t), with the initial condition that production was 100 cars at t=0 (initial time), and it increased to 200 cars after t=10 years. To solve this, we integrate to find C(t) = αe^{kt}. We can determine the values of α and k using the initial condition and the given production value after 10 years.

Applying the initial condition C(0) = αe^{k*0} = 100, we find that α = 100. Next, using the condition C(10) = αe^{k*10} = 200, we can solve for k by substituting α and simplifying the equation: 200 = 100e^{k*10}, which upon further simplification gives us k = ln(2)/10. Hence, we have C(t) = 100e^{(ln(2)/10)t}.