5. suppose that you have 600 inches of fencing to make two adjacent rectangular kennels of length x inches…

Explanation:

Step1: Set up the perimeter equation

The total length of the fencing is (3x + 2y=600). We can express (y) in terms of (x) as (y = 300-\frac{3}{2}x).

Step2: Set up the area function

The area (A) of the two - adjacent rectangular kennels is (A=2xy). Substitute (y = 300-\frac{3}{2}x) into the area formula: (A(x)=2x\left(300-\frac{3}{2}x\right)=600x - 3x^{2})

Step3: Find the first derivative of the area function

Using the power rule ((x^{n})^\prime=nx^{n - 1}), we have (A^\prime(x)=600-6x)

Step4: Find the critical points

Set (A^\prime(x) = 0), then (600-6x=0). Solving for (x): (6x=600), so (x = 100) inches.

Step5: Find the second derivative of the area function

Differentiate (A^\prime(x)) with respect to (x). (A^{\prime\prime}(x)=-6) Since (A^{\prime\prime}(x)=-6<0) for all (x), the function (A(x)) is concave down for all (x).

Step6: Find the value of (y)

Substitute (x = 100) into (y = 300-\frac{3}{2}x) (y=300-\frac{3}{2}\times100=300 - 150=150) inches

Answer:

The value of (x) is (100) inches and the value of (y) is (150) inches. Since the second - derivative (A^{\prime\prime}(x)=-6<0), the function (A(x)) is concave down, and the critical point (x = 100) (and corresponding (y = 150)) gives an absolute maximum for the area function.