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: Find value after 3 months

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

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.
$m>\frac{\ln(\frac{10}{19})}{\ln(0.92)}=\frac{\ln(10)-\ln(19)}{\ln(0.92)}\approx\frac{2.3026 - 2.9444}{- 0.0834}=\frac{- 0.6418}{- 0.0834}\approx7.7$.
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.