Question

A bicycle loses its value each month after it is purchased. Its value as a function of time, in months, is modelled by /) - 2200(0 98)*. In which month after it is purchased does the bicycle's worth fall below $1500?

Answer 1

To find the month after which the bicycle’s worth falls below $1500, we start with the value function given by:

\[
V(t) = 2200(0.98)^t
\]

We need to determine when \( V(t) < 1500 \). We can set up the inequality:

\[
2200(0.98)^t < 1500
\]

To solve for \( t \), we first divide both sides of the inequality by 2200:

\[
(0.98)^t < \frac{1500}{2200}
\]

Calculating the right side:

\[
\frac{1500}{2200} = \frac{15}{22} \approx 0.681818
\]

Now we have:

\[
(0.98)^t < 0.681818
\]

Next, we take the logarithm of both sides to solve for \( t \):

\[
\log((0.98)^t) < \log(0.681818)
\]

Using the properties of logarithms:

\[
t \cdot \log(0.98) < \log(0.681818)
\]

Since \( \log(0.98) \) is negative, we divide by \( \log(0.98) \) and reverse the inequality:

\[
t > \frac{\log(0.681818)}{\log(0.98)}
\]

Now we can calculate the values of the logarithms:

\[
\log(0.681818) \approx -0.169
\]
\[
\log(0.98) \approx -0.0087
\]

Now substitute these values into the right-hand side:

\[
t > \frac{-0.169}{-0.0087} \approx 19.4
\]

Since \( t \) must be a whole month, we round up to the nearest whole number. Therefore, the bicycle's worth falls below $1500 in:

\[
t = 20
\]

Thus, the bicycle's worth falls below $1500 after **20 months**.