Question

Newton's law of cooling indicates that the temperature of a warm object will decrease exponentially with time and will approach the temperature of the surrounding air. The temperature T(t) is modeled by T(t)=Tₐ+(T₀-Tₐ)e⁻ᵏᵗ. In this model, Tₐ represents the temperature of the surrounding air, T₀ represents the initial temperature of the object, tis the time after the object starts cooling, and K is the cooling rate. Water in a water heater is originally 131°F. The water heater is shut off, and the water cools to the temperature of the surrounding air, which is 80°F. The rate of cooling is 0.00348. Dominic does not like to shower with water less than 116°F. If Dominic waits 24 hours after the heater is shut off, will the water still be warm enough for a shower? How warm will the water be?

a) Yes, the water will be warm enough.
b) No, the water will not be warm enough.
c) Insufficient information to determine.
d) The temperature cannot be calculated.

Answer 1

Using Newton's law of cooling, the water temperature after 24 hours is calculated to be approximately 126.935°F, which is above Dominic's minimum shower temperature requirement of 116°F. Hence, the water will still be warm enough for a shower.

Step-by-step explanation:

To determine if the water in the water heater will still be warm enough for Dominic to shower after 24 hours, we can use Newton's law of cooling. Given that the initial temperature of the water (T₀) is 131°F, the surrounding air temperature (Tᵢ) is 80°F, the cooling rate (k) is 0.00348, and the time (t) is 24 hours, we can plug these values into the temperature model T(t) = Tᵢ + (T₀-Tᵢ)e−kt to calculate the water temperature after 24 hours.

Using the equation:

T(24) = 80 + (131 - 80)e−(0.00348)(24)

We find that the temperature of the water after 24 hours is:

T(24) = 80 + 51e−(0.08352)

T(24) ≈ 80 + 51(0.9199)

T(24) ≈ 80 + 46.935

T(24) ≈ 126.935°F

Since 126.935°F is greater than Dominic's minimum shower temperature of 116°F, the answer is:

a) Yes, the water will be warm enough.