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6th Edition

Stewart + 5 others

Publisher: Cengage Learning

ISBN: 9780840068071

Chapter 2, Problem 23P

(a)

To determine

**To find:** The function that models the cost of fencing of the garden.

Expert Solution

The function that models the cost of fencing of the garden is

**Formula used:**

The area of the rectangle is,

Perimeter of the rectangle is,

**Given:**

The cost of fence next to the road is

The total area of the garden is

**Calculation:**

Let the length of the rectangular garden is *x* units and the breadth of the rectangular garden is *y* units.

Substitute *x* for length and *y* for breadth in equation (1),

Now, substitute *x* for length and *y* for breadth in equation (2),

Then, the cost of fencing is,

Substitute *y* in equation (4),

Thus, the function that models the cost of fence of the garden is

(b)

To determine

**To find:** The garden dimensions that minimize the cost of fence.

Expert Solution

The length of the garden that minimize the cost of fence is

**Calculation:**

To find the minimum cost of the fence of the rectangular garden, the graph of the function

The function contains the variable *x* as the length of the rectangular garden.

The local minimum value of the function is the least finite value where the value of the function at the any number is less than to the original function.

The condition for local minimum is,

The graph of the function is shown below,

From the above Figure, it can be observed that the least peak is at the point

Then the minimum cost is

Substitute

Thus, the length of the rectangular garden is 30 units and the breadth is 40 units.

(c)

To determine

**To find:** The range of length that can be fenced along the road with an amount of $600.

Expert Solution

The range of the length of the rectangular garden that owner fence along the road is

**Calculation:**

From the part (a), cost function

If the owner has at most

Substitute

Take the equality of the equation (5),

Further solving,

Thus, the range of the length is between