HW6 Solutions - ASSIGNMENT 6 SOLUTIONS Problem 1 Here is a...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: ASSIGNMENT 6 - SOLUTIONS Problem 1. Here is a guesstimate. In both parts, we are supposed to find a distinguished rectangle among all rectangles inscribed in a given circle. For reasons of symmetry, the only distinguished rectangle that comes to mind is a square. Heuristically, we expect a square to be the optimal shape for both a) and b). Now let’s check using calculus. Call x and y the dimensions of the rectangle. Since both diagonals are diameters, we have the Pythagorean relation x 2 + y 2 = (2 r ) 2 = 4 r 2 , hence y = √ 4 r 2- x 2 . a) We have to maximize the perimeter P = 2 x + 2 y . After substituting y , we get to the function P ( x ) = 2( x + ð 4 r 2- x 2 ) with feasible domain (0 , 2 r ). The derivative is P Í ( x ) = 2 1- x √ 4 r 2- x 2 ; we find the critical numbers from P Í ( x ) = 0 or P Í ( x ) DNE. The first gives x = r √ 2; the second has no solution within the feasible domain. To check that the critical number x = r √ 2 is indeed a point where P ( x ) achieves its absolute maximum, we look at the sign of P Í ( x ). Between 0 and r √ 2, P Í ( x ) is positive hence P ( x ) is increasing. Between r √ 2 and r , P Í ( x ) is negative hence P ( x ) is decreasing. We conclude that P ( r √ 2) = 4 r √ 2 is the largest possible perimeter, and it is realized by the square with side-length r √ 2....
View Full Document

This note was uploaded on 07/31/2011 for the course MATH 102 taught by Professor Maryamnamazi during the Spring '10 term at University of Victoria.

Page1 / 3

HW6 Solutions - ASSIGNMENT 6 SOLUTIONS Problem 1 Here is a...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online