A 20-year-old university student weighs 138 lb and had a birth weight of 6 lb. Prove that at some point in her life she was growing at a rate of 6.6 pounds per year.

Solution

VerifiedWe are told that a university student weighs 138 lbs currently and at the time of her birth she weighed 6 lbs.

The average rate of increment of her weight is

$W_{avg} = \dfrac{138-6}{20-0} = 6.6 \text{ pounds/year}$

Now the Mean Value theorem tells us that there exists a $c$ between $a$ and $b$, such that

$f'(c) = \dfrac{f(b)-f(a)}{b-a}$

In this case, we have the weight at 20 years as 138 lbs and at 0 years(approximately 1 day) as 6 lbs. Hence using the mean value theorem we should have a $c$ such that

$f'(c) = \dfrac{138-6}{20-0} = 6.6 \text{ pounds/year}$

Hence this tells us that at some point the average increment shall be equal to the instantaneous increment of weight.