Question

A tire in Manny's car has just sprung a leak and is losing pressure at a rate of 4% every hour. If the tire pressure is currently 248 kilopascals, what will it be in 20 hours?

Answer 1

To find the tire pressure after 20 hours with a leak rate of 4% per hour, we can use the formula for exponential decay:

\[ P(t) = P_0 \times (1 - r)^t \]

where:
- \( P(t) \) is the pressure after time \( t \),
- \( P_0 \) is the initial pressure,
- \( r \) is the decay rate (as a decimal), and
- \( t \) is the time in hours.

Given:
- \( P_0 = 248 \) kPa
- \( r = 4\% = 0.04 \)
- \( t = 20 \) hours

Now, we can plug these values into the equation:

\[
P(20) = 248 \times (1 - 0.04)^{20}
\]

Calculating \( (1 - 0.04) \):

\[
1 - 0.04 = 0.96
\]

Now calculate \( 0.96^{20} \):

\[
0.96^{20} \approx 0.6676 \quad (\text{using a calculator})
\]

Now we can find \( P(20) \):

\[
P(20) = 248 \times 0.6676 \approx 165.5
\]

Thus, the tire pressure after 20 hours will be approximately **165.5 kilopascals**.