Question

part 3 of 5
(d) express the equation needed for derivation in terms of \\(x\\): \\(a(x) = \frac{740x - 8x^2}{3}\\)
(e) \\(a(x) = \frac{740 - 16x}{3}\\)
part 4 of 5
(f) the critical value(s) in the domain is \\(46.25\\)
(g) use the second derivative test to ensure the critical value gives a maximum value for the area
\\(a(x) = -\frac{16}{3}\\) and \\(a(cv) = -\frac{16}{3}\\)
since \\(a(cv)\\) is negative, the critical value is at a maximum
continue to the next part.
part 5 of 5
(h) by the select an answer, the maximum area will have an \\(x\\) dimension of feet by a \\(y\\) dimension of feet using 370 feet of fencing material.

Explanation:

Identify the optimization context

Using the Optimization Problems and Domain of Optimization knowledge points

$$
LATEXBLOCK0
$$

Determine the applicable theorem

Using the Optimization Problems knowledge point

$$
LATEXBLOCK1
$$

Calculate the optimal dimensions

Using the Critical Points and Optimization Problems knowledge points

$$
LATEXBLOCK2
$$

Answer:

By the Extreme Value Theorem, the maximum area will have an \(x\) dimension of 46.25 feet by a \(y\) dimension of 61.67 feet using 370 feet of fencing material.