To find: The function that models the cost of fencing of the garden.
The function that models the cost of fencing of the garden is .
The area of the rectangle is,
Perimeter of the rectangle is,
The cost of fence next to the road is per foot and the cost of fence of other boundaries is per foot.
The total area of the garden is .
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 for y in equation (4),
Thus, the function that models the cost of fence of the garden is .
To find: The garden dimensions that minimize the cost of fence.
The length of the garden that minimize the cost of fence is and the breadth is .
To find the minimum cost of the fence of the rectangular garden, the graph of the function has to be drawn.
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 at .
Substitute in equation (3),
Thus, the length of the rectangular garden is 30 units and the breadth is 40 units.
To find: The range of length that can be fenced along the road with an amount of $600.
The range of the length of the rectangular garden that owner fence along the road is .
From the part (a), cost function .
If the owner has at most then the cost function is less than or equal to .
Substitute for in cost function ,
Take the equality of the equation (5),
Thus, the range of the length is between to fence the length of rectangular garden if the owner has at most .
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