Question

an amusement company maintains records for each video game it installs in an arcade. suppose that ( c(t) ) and ( r(t) ) represent the total accumulated costs and revenues (in thousands of dollars), respectively, ( t ) years after a particular game has been installed and the derivatives ( c^{prime}(t) ) and ( r^{prime}(t) ) are the following functions.
( c^{prime}(t)=2 )
and
( r^{prime}(t)=7 e^{-0.4 t} )
find the area between the graphs of ( c^{prime} ) and ( r^{prime} ) over the interval on the ( t )-axis from 0 to the useful life of the game and interpret the results.
what is the useful life of the game?
( t=square ) years (round to the nearest tenth as needed.)

Explanation:

Step1: Set \(C^{\prime}(t)=R^{\prime}(t)\)

Set \(2 = 7e^{-0.4t}\).

Step2: Solve for \(t\)

First, rewrite the equation as \(e^{-0.4t}=\frac{2}{7}\).
Then, take the natural - logarithm of both sides: \(\ln(e^{-0.4t})=\ln(\frac{2}{7})\).
Using the property \(\ln(e^{x}) = x\), we get \(-0.4t=\ln(\frac{2}{7})\).
Since \(\ln(\frac{2}{7})=\ln(2)-\ln(7)\approx0.693 - 1.946=-1.253\).
So, \(t=\frac{\ln(\frac{2}{7})}{- 0.4}=\frac{-1.253}{-0.4}=3.1325\approx3.1\).

Answer:

\(t = 3.1\) years