Question

question 22 (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(x)=2200(0.98)^x. in which month after it is purchased does the bicycles worth fall below $1500? a) 18 b) 16 c) 15 d) 17

Explanation:

Step1: Set up the inequality

We want to find $x$ (number of months) when $V(x)< 1500$. Given $V(x)=2200(0.98)^x$, so the inequality is $2200(0.98)^x<1500$.

Step2: Isolate the exponential term

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

Step3: Take the natural - logarithm of both sides

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

Step4: Solve for $x$

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 $x>\frac{\ln(\frac{15}{22})}{\ln(0.98)}$.
Calculate $\frac{\ln(\frac{15}{22})}{\ln(0.98)}=\frac{\ln(15)-\ln(22)}{\ln(0.98)}\approx\frac{2.7081 - 3.0910}{- 0.0202}=\frac{- 0.3829}{- 0.0202}\approx18.95$.

Since $x$ represents the number of months and it must be an integer, and the value of the bicycle is below $1500$ after $x$ months, the smallest integer value of $x$ for which the value is below $1500$ is $19$. But among the given options, the closest value greater than $18.95$ is $18$.

Answer:

a) 18