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 5. The sum of two positive numbers is 30. Find the numbers if the sum of their squares is a minimum. Important Notes: 1) We can denote the problem as x + y = 30, wherein we will minimize x2+ y2 . [Where x is the first number and y is the second number]. 2) Since x+ y=30, then y=30−x . a. What is the objective? Let it be 𝑆(𝑥). The objective is the sum of the squares of the two positive numbers. We are required to find the numbers that will minimize S. b. What variable are you going to control? Let it be 𝑥. Let x be the first number. Let it be the control variable. Let 30−x be the second number (Notice that y is the second number on “Important Notes” number 1). c. What function accurately models this problem? Our model is S ( x )=x2+ (30−x ) 2 =x2+900−60 x+x2=2 x2−60x+900. Let it be continuous over [0, 30]. d. What are the numbers? Finding its critical points. S' (x )=(2 )2 x2−1−60 x1−1+900=4 x−60 4 x−60=0→x=15 Evaluating P at 0, 15, 30 x 0 15 30 S(x) 900 Max 450 Min 900 Max To minimize S, the value of the first number should be 15 while the value of the first number should be (30−15)¿15 Therefore, the first and second number are 15 and 15. e. What is the minimum sum of their squares? The minimum sum of the squares of the first number and the second number is P (15 )=2(15)2−60 (15 )+900=2 (225 )−900+900=450Alternative Method: Using x2+ y2 x2+ y2=(15)2+(15)2=225+225=450