Basic Calculus - Problem 2 on SOLVING OPTIMIZATION PROBLEMS USING CALCULUS, Exercises of Mathematics

Download Basic Calculus - Problem 2 on SOLVING OPTIMIZATION PROBLEMS USING CALCULUS and more Exercises Mathematics in PDF only on Docsity! Basic Calculus Performance Task 2 SOLVING OPTIMIZATION PROBLEMS USING CALCULUS 2. A rectangular chicken pen is bounded on one side by the wall of the house and the other three sides by 160 meters of fences. Find the dimensions of the chicken pen if the area is a maximum. a. What is the objective? Let it be 𝐴(𝑥). The objective is the area of rectangular chicken. We are required to find the dimensions that give the biggest area. b. Sketch the chicken pen. Label the important parts. c. What variable are you going to control? Let it be 𝑥. As the length of the wall is fixed, let the fixed value be y and the variable we are going to control be x. Let x be the width of the chicken pen. d. What function accurately models this problem? Given that the total length of the fence is 160m and with the aid of the illustration, We have, 2 x+ y=160→ y=160−2x Now, the question is to find the dimensions of the chicken pen if the area is maximum. Therefore, the area of the rectangle pen Area, A=xy would be A=x (160−2x )=160 x−2 x2. Let it be continuous over [0, 80]. Our model would be A(x )=160 x−2 x2 e. What are the dimensions of the chicken pen? Before finding the critical points, we must first find the derivative of the function and equate it to zero and find the value of x. A ( x )=160 x−2 x2 A ' (x )=(1 )160x1−1−(2 )2 x2−1=160−4 x 160−4 x=0→x=40 Evaluating 𝐴 at 0, 40, and 80. x 0 40 80 y Fence x FenceFence x Wall of the house Chicken pen