Question

question 4 (mandatory) (1 point)
a bicycle loses its value each month after it is purchased. its value as a function of time, in months, is modelled by $v(m)=2200(0.98)^m$. in which month after it is purchased does the bicycle’s worth fall below $1500?
a) 17
b) 15
c) 18
d) 16

Explanation:

Step1: Set up the inequality

We want to find $m$ such that $V(m)=2200(0.98)^m < 1500$. So, we have the inequality $2200(0.98)^m<1500$. First, divide both sides by 2200: $(0.98)^m<\frac{1500}{2200}=\frac{15}{22}$.

Step2: Take the natural - logarithm of both sides

$\ln(0.98)^m<\ln(\frac{15}{22})$. Using the property of logarithms $\ln(a^b) = b\ln(a)$, we get $m\ln(0.98)<\ln(\frac{15}{22})$.

Step3: Solve for $m$

Since $\ln(0.98)<0$, when we divide both sides of the inequality $m\ln(0.98)<\ln(\frac{15}{22})$ by $\ln(0.98)$, the direction of the inequality sign changes. So, $m>\frac{\ln(\frac{15}{22})}{\ln(0.98)}$.
We know that $\ln(\frac{15}{22})\approx\ln(15)-\ln(22)\approx2.708 - 3.091=- 0.383$ and $\ln(0.98)\approx - 0.0202$. Then $m>\frac{-0.383}{-0.0202}\approx18.96$.

Answer:

c) 18