Question

The relation T=190(1/2)^1/10 can be used to determine the length of time, t, in hours, that milk of a certain fat content will remain fresh. T is the storage temperature, in degrees Celsius.

A.) What is the freshness half life of milk?
B.) How long will milk keep fresh at 4 degrees Celsius?

Answer 1

Are you sure that isn't \[T = 190(\frac{1}{2})^{t/10}\]or something similar? The expression you have given us does not use \(t\) at all, and has a fixed value.

Exponential decay relationships of this sort (using a power of \(1/2\)) can be easily deciphered. The number in the divisor of the exponent is the half life. At time \(t=0\), \[(\frac{1}{2})^{t/\text{half-life}} = (\frac{1}{2})^0 = 1\]meaning that all of the substance is still present.

At time \(t = \text{half-life}\), you have

\[(\frac{1}{2})^{\text{half-life}/\text{half-life}} = (\frac{1}{2})^1 = \frac{1}{2}\]indicating that half of the substance is no longer present, thus the term "half-life".

To find out how long the milk will stay fresh at some temperature \(T\), you need to rearrange the equation to solve for \(t\). Do you remember your properties of logarithms and exponentials?