According to Newton’s law of cooling, the time t needed for an object at an initial temperature T0 to cool to a temperature T in an environment with ambient temperature Ta is given by

where k is a constant. Assume that for a certain type of container, k = 0.025 min-1. Let t be the number of minutes needed to cool the container to a temperature of 50°F. Assume that T0 = 70.1 ± 0.2°F and Ta = 35.7 ± 0.1 °F. Estimate t, and find the uncertainty in the estimate.

Answer:

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In this question we are asked to find the estimate of Newton’s law cooling equation and the uncertainty in the estimate.

Newton’s law cooling equation = -…………..(1)

Where

t is the number of minutes needed to cool the container

is a initial temperature

is an ambient temperature

is a constant

is a cooling temperature

Given parameters:

= 0.025 , = 70.1 ± 0.2°F, = 35.7 ± 0.1 °F and

The quantity which we are estimating has several measurements. In this question we are measuring the initial temperature and ambient temperature . These quantity mentioned here is uncertain and independent measurements.

So here we can use the following basic formula.

……………...(2)

Where , ,........, are independent measurements and uncertainties are , ,.........,

Equation 2 is called the multivariate propagation of error formula.