$\"\"$

(a).

The amount of drinking water reserves in millions of gallons available for a town is modeled by $\"\"$ .

The minimum amount of water needed by the residents is modeled by $\"\"$ .

Where $\"\"$ is time in years.

$\"\"$ is a polynomial function.

$\"\"$ is a power or radical function.

$\"\"$

(b).

Since time and the amount of water cannot be negative, the relevant domains for $\"\"$ and $\"\"$ are restricted to nonnegative values for $\"\"$ that result in nonnegative values for $\"\"$ and $\"\"$ .

Draw the coordinate plane.

Graph the functions $\"\"$ and $\"\"$ on the same graph.

Graph :

$\"\"$

Observe the graph :

The function $\"\"$ cuts the $\"\"$ -axis at $\"\"$ .

Therefore the domain for $\"\"$ is $\"\"$ .

For the range, find the maximum using $\"\"$ .

$\"\"$

The vertex occurs when $\"\"$

Subustiute the value of $\"\"$ in $\"\"$ to find the range.

$\"\"$

Therefore the range for $\"\"$ is $\"\"$ .

The domain of $\"\"$ is $\"\"$ .

The range of $\"\"$ is $\"\"$ .

$\"\"$

(c).

Graph both functions.

The domain of $\"\"$ is $\"\"$ , so analyze $\"\"$ as $\"\"$ approaches both $\"\"$ and $\"\"$ .

Draw the coordinate plane.

Graph the functions $\"\"$ and $\"\"$ on the same graph.

Graph :

$\"\"$

Observe the graph :

For $\"\"$ , as $\"\"$ approaches $\"\"$ , $\"\"$ approaches $\"\"$ . As $\"\"$ approaches $\"\"$ , $\"\"$ approaches $\"\"$ .

For $\"\"$ , as $\"\"$ approaches $\"\"$ , $\"\"$ approaches $\"\"$ . As $\"\"$ approaches $\"\"$ , $\"\"$ also approaches $\"\"$ .

$\"\"$

(d).

Draw the coordinate plane.

Graph the functions $\"\"$ and $\"\"$ on the same graph.

Graph :

The graph of $\"\"$ and $\"\"$ when $\"\"$ .

$\"\"$

(e).

$\"\"$ is a polynomial function, so it is also continuous and the Intermediate Value Theorem applies.

Therefore, since $\"\"$ and $\"\"$ , it follows that there is a number $\"\"$ , such that $\"\"$ and $\"\"$ .

$\"\"$

(f).

The domain of $\"\"$ is $\"\"$

When $\"\"$ , $\"\"$ .

This means that the town will run out of water reserves after about $\"\"$ years.

$\"\"$

(g).

Draw the coordinate plane.

Graph the functions $\"\"$ and $\"\"$ on the same graph.

Find the point at which both the graphs are intersected.

Graph :

$\"\"$

Observe the graph :

The functions $\"\"$ and $\"\"$ get intersects at $\"\"$ .

$\"\"$ .Therefore $\"\"$ .

In about $\"\"$ years, the residents will need more water than what is in their reserves.

$\"\"$

(a).

$\"\"$ is a polynomial function.

$\"\"$ is a power or radical function.

(b).

The Domain of $\"\"$ is $\"\"$ ,

The Range of $\"\"$ is $\"\"$ .

The Domain of $\"\"$ is $\"\"$ .

The Range of $\"\"$ is $\"\"$ .

(c).

Graph :

$\"\"$

For $\"\"$ , as $\"\"$ approaches $\"\"$ , $\"\"$ approaches $\"\"$ . As $\"\"$ approaches $\"\"$ , $\"\"$ approaches $\"\"$ .

For $\"\"$ , as $\"\"$ approaches $\"\"$ , $\"\"$ approaches $\"\"$ . As $\"\"$ approaches $\"\"$ , $\"\"$ also approaches $\"\"$ .

(d).

Graph :

The graph of $\"\"$ and $\"\"$ when $\"\"$ .

$\"\"$

(e).

Since $\"\"$ and $\"\"$ , it follows that there is a number $\"\"$ , such that $\"\"$ and $\"\"$ .

(f).

This means that the town will run out of water reserves after about $\"\"$ years.

(g).

Graph :

$\"\"$

In about $\"\"$ years, the residents will need more water than what is in their reserves.