starting with a 140 - foot - long stone wall, a farmer would like to construct a rectangular enclosure by…
starting with a 140 - foot - long stone wall, a farmer would like to construct a rectangular enclosure by adding 980 feet of fencing, as shown in the figure to the right. find the values of x and w that result in the greatest possible area. x = ft w = ft
Answer
Explanation:
Step1: Set up the perimeter equation
The total length of the fencing used is (2x + w=980) (using the existing 140 - foot - long stone wall), so (w = 980 - 2x).
Step2: Set up the area function
The area (A) of a rectangle is (A=x\times w). Substitute (w = 980 - 2x) into the area formula, we get (A(x)=x(980 - 2x)=980x-2x^{2}).
Step3: Find the derivative of the area function
The derivative (A'(x)=\frac{d}{dx}(980x - 2x^{2})=980-4x).
Step4: Set the derivative equal to zero
Set (A'(x) = 0), so (980-4x = 0). Solving for (x): [ \begin{align*} 980-4x&=0\ 4x&=980\ x& = 245 \end{align*} ]
Step5: Find the value of (w)
Substitute (x = 245) into the equation (w=980 - 2x). Then (w=980-2\times245=980 - 490 = 490).
Answer:
(x = 245) ft, (w = 490) ft