Q. 76 - Calculus

Question

Explain in practical terms what the Extreme Value Theorem says about each continuous function defined in Exercises $75-77$. Then explain in practical terms what the Intermediate Value Theorem says in each situation. 

Linda collects rain in a bucket outside her back door. Since the first day of April she has been keeping track of how the amount of water in the bucket changes as it fills with rain and evaporates. On April $1$ the bucket was empty, and today it contains $4$ inches of water. Let $w\left(t\right)$ be the height, in inches, of rainwater in the bucket $t$ days after the first day of April.

Step-by-Step Solution

Answer

The Extreme Value Theorem tells us that $w\left(0\right)=0$ initially and $w\left(t\right)=4$ after $t$ days.

The Intermediate Value Theorem tells us that for every day between April $1$ and today's date the height of water is somewhere from $0$ inches to $4$ inches.  

Step 1. Given Information

Height of water in bucket on April $1=$empty

Height of water in bucket now $=4$ inches

It is given that $w\left(t\right)$ be the height, in inches, of rainwater in the bucket $t$ days after the first day of April.

We need to find what the Extreme Value Theorem and Intermediate Value Theorem says in this situation. 

Step 2. Extreme Value Theorem

The height function $w\left(t\right)$ should be continuous because the height of water in the bucket changes continuously over time and cannot jump from one value to another.

The Extreme Value Theorem tells us that $w\left(0\right)=0$ initially. On April $1$ the height is $0$ inches.

On the other hand, the height is $4$ inches after $t$ days.

$\therefore w\left(t\right)=4$

Step 3. Intermediate Value Theorem

The Intermediate Value Theorem tells us that for every day between April $1$ and today's date the height of water is somewhere from $0$ inches to $4$ inches.