Solving Optimization Problems using Calculus, Exercises of Mathematics

Download Solving Optimization Problems using Calculus and more Exercises Mathematics in PDF only on Docsity! Basic Calculus Performance Task 2 SOLVING OPTIMIZATION PROBLEMS USING CALCULUS 1) Find two positive numbers whose sum is 20 if the product of the first number and the square of the second number is to be a maximum. Important Notes: 1) We can denote the problem as x + y = 20, wherein we will maximize x y2. 2) Since x+ y=20, then y=20−x . a. What is the objective? Let it be 𝑃(𝑥). The objective is the product of the first number and the square of the second number. We are required to find the numbers that will maximize 𝑃. b. What variable are you going to control? Let it be 𝑥. Let x be the second number. Let it be the control variable. Let 20−x be the first number. c. What function accurately models this problem? Our model is P ( x )=(20−x ) x2=20 x2−x3 . Let it be continuous [0,20 ] d. What are the numbers? Finding its critical points. P '(x )=(2 )20x2−1−(3 ) x3−1=40 x−3 x2 40 x−3 x2=0→x=0 , 40 3 Evaluating P at 0, 40 3 , 20 x 0 40 3 20 P(x) 0 Min 32000 27 Max 0 Min To maximize 𝑃, the value of the second number should be 40 3 while the value of the first number should be (20−403 )=60−403 = 20 3 Therefore, the first and second number are 20 3 and 40 3 , respectively.