Question

question 15 (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 bicycles worth fall below $1500? a) 16 b) 18 c) 17 d) 15

Explanation:

Step1: Set up the inequality

We want to find when $V(m)=2200(0.98)^m < 1500$. So we set up the inequality $2200(0.98)^m<1500$.

Step2: Isolate the exponential term

Divide both sides of the inequality by 2200: $(0.98)^m<\frac{1500}{2200}=\frac{15}{22}$.

Step3: 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 $m\ln(0.98)<\ln(\frac{15}{22})$.

Step4: Solve for m

Since $\ln(0.98)<0$, when we divide both sides of the inequality 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 $m>\frac{-0.383}{-0.0202}\approx18.96$.

Since $m$ represents the number of months and it must be an integer, the smallest value of $m$ for which the bicycle's worth falls below $1500$ is $m = 19$. But among the given options, the closest value greater than $18.96$ is $18$.

Answer:

b) 18