Question

farmer hannah has 400 feet of fencing to enclose 2 adjacent horse corrals, as shown in the diagram above.
what dimensions should be used so that the enclosed area will be maximized?

(a) express the area \\(a\\) as an equation of \\(x\\) and \\(y\\): \\(a = 2xy\\)
(b) determine the perimeter equation as a function of \\(x\\) and \\(y\\) with the given information:
\\(4x + 3y = 400\\)

(c) determine the domain for the given problem in terms of \\(x\\):

Explanation:

Formulate the constraint equation

Using the Interval Notation knowledge point

$$ 4x + 3y = 400 $$

Determine physical boundaries for variables

To find the domain in terms of \(x\), we analyze the physical constraints on the dimensions:

• The width \(x\) must be strictly positive: \(x > 0\).
• The height \(y\) must be strictly positive: \(y > 0\).

Express y in terms of x

Using the constraint equation, we solve for \(y\):

$$ 3y = 400 - 4x \implies y = \frac{400 - 4x}{3} $$

Solve the inequality for x

Since \(y > 0\), we set up and solve the inequality:

$$ \frac{400 - 4x}{3} > 0 \implies 400 - 4x > 0 \implies 4x < 400 \implies x < 100 $$

Express the domain in interval notation

Combining the physical constraints \(x > 0\) and \(x < 100\), we obtain the open interval:

$$ (0, 100) $$

Answer:

(C) Determine the domain for the given problem in terms of \(x\): <blank>\((0, 100)\)</blank>