2. Ailim is a muralist who creates her murals by assembling irregularly-shaped tiles2 . Anna, an art enthusiast and a tile collector, notices one of Ailim's works at the Eden Gallery. Enamored by its perfection, Anna wishes to recreate the masterpiece at home. Anna designs each tile of Ailim's mural on her computer, and recreates it at her friend's workshop. Anna uploads several pictures of the mural to her computer, and she has a program that attempts to outline each tile using the photos. The program works remarkably well for most of the mural, but the software encounters difficulties in a small section of the mural with minuscule tiles. The software models top edge of each minuscule tile by a part of the graph of the function E(x) = ? 1 ? x 2. The sides and bottom edges of the minuscule tiles appear to be essentially straight. Anna uses this information, along with area results from her sensitive planimeter (a device used to measure the area of two dimensional shapes), to accurately outline these tiles. According to the software model, each minuscule tile has area3

A(t) = integral from ?2t to 1-t E(x) dx = integral from ?2t to 1-t sqrt(1 ? x^2) dx, for some number t, where 0 ? t ? 1 2 . The units of t and x are millimeters, and the units of A(t) are square millimeters. Note that, for any fixed t, the following outline for the (minuscule) tile is implicitly defined by the expression for A(t).

Use computational software to find the value of any integrals you need for either part of the problem. 

(a) Anna finds that her planimeter outputs the area of one of the tiles to be just slightly over the maximum area allowed by A. Instead of using the area given by the planimeter, she decides to use the area given by the maximum of A(t). For what value of t does A(t) attain its maximum? What are the values of xleft and xright, as defined in Figure 1, when A(t) is at a maximum? 

(b) Anna's friend, Brian, owns a workshop and specializes in cutting tiles. Brian's machinery can accurately cut tiles that have area larger than 3 4 mm^2 . If Anna presents Brian with a tile outline output from the software model, is it possible that Brian won't be able to cut it?

Transcribed Image Text:

Cleft = -2t E(x) = 1 x right = 1 - t Figure 1: Outline of a (minuscule) tile with area given by A(t) Cleft = -2t E(x) = 1 x right = 1 - t Figure 1: Outline of a (minuscule) tile with area given by A(t)