A potato is placed in a preheated oven to bake. Its temperature \( p = p(t) \) is given by:

\[ p = 400 - 325e^{-t/50} \]

where \( p \) is measured in degrees Fahrenheit and \( t \) is the time in minutes since the potato was placed in the oven.

a) What is the initial temperature of the potato when placed in the oven?

b) How does the temperature change over time in the oven?

c) What is the temperature of the potato after 30 minutes in the oven?

d) At what time will the potato reach a temperature of 200 degrees Fahrenheit?

The initial temperature of the potato is 75 degrees Fahrenheit. Its temperature approaches 400 degrees Fahrenheit over time, reaching 254 degrees after 30 minutes. The potato will reach 200 degrees Fahrenheit in approximately 25.54 minutes.

Explanation:

To determine the initial temperature of the potato, we set t = 0 in the equation p = 400 - 325e(-t/50). This yields: p(0) = 400 - 325e(0) = 400 - 325(1) = 75 degrees Fahrenheit. The temperature change over time is dictated by the decay term -325e(-t/50), which indicates that as t increases, the temperature of the potato approaches 400 degrees Fahrenheit asymptotically.

To find the temperature after 30 minutes: p(30) = 400 - 325e(-30/50) ⇒ p(30) ≈ 400 - 325(0.4493) ≈ 400 - 146 ≈ 254 degrees Fahrenheit. For the potato to reach 200 degrees Fahrenheit, we set p(t) = 200 and solve for t: 200 = 400 - 325e(-t/50), 200 - 400 = - 325e(-t/50), -200 = -325e(-t/50), e(-t/50) = 200/325, -t/50 = ln(200/325), t = -50 * ln(200/325), t ≈ -50 * (-0.5108) ≈ 25.54 minutes.