Question

question 14 (mandatory) (1 point)
a laptop computer loses its value each month after it is purchased. its value as a function of time, in months, is modelled by $v(m)=3800(0.92)^m$. what is the value of the laptop after 3 months? in which month after it is purchased does the laptops worth fall below $2000?

Explanation:

Step1: Calculate value after 3 months

Substitute $m = 3$ into $V(m)=3800(0.92)^m$.
$V(3)=3800\times(0.92)^3$
$V(3)=3800\times0.778688$
$V(3)=2958.0144\approx2958.01$

Step2: Find month when value falls below $2000$

Set $V(m)<2000$, so $3800(0.92)^m<2000$.
First, divide both sides by 3800: $(0.92)^m<\frac{2000}{3800}=\frac{10}{19}$.
Take the natural - logarithm of both sides: $m\ln(0.92)<\ln(\frac{10}{19})$.
Since $\ln(0.92)<0$, when dividing by $\ln(0.92)$ the inequality sign flips.
$m>\frac{\ln(\frac{10}{19})}{\ln(0.92)}=\frac{\ln(10)-\ln(19)}{\ln(0.92)}$.
$\ln(10)\approx2.3026$, $\ln(19)\approx2.9444$, $\ln(0.92)\approx - 0.0833$.
$m>\frac{2.3026 - 2.9444}{-0.0833}=\frac{-0.6418}{-0.0833}\approx7.71$.
Since $m$ represents the number of months and it must be an integer, $m = 8$.

Answer:

The value of the laptop after 3 months is approximately $\$2958.01$. The laptop's worth falls below $\$2000$ in the 8th month.