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An indoor physical fitness room consists of a rectangular region with a semicircle on each end. The perimeter of the room is to be a 200-meter single-lane running track.

(a) Draw a diagram that illustrates the problem. Let $ x $ and $ y $ represent the length and width of the rectangular region, respectively.(b) Determine the radius of each semicircular end of the room. Determine the distance, in terms of $ y $ around the inside edge of each semicircular part of the track.(c) Use the result of part (b) to write an equation, in terms of $ x $ and $ y $, for the distance traveled in one lap around the track. Solve for $ y $.(d) Use the result of part (c) to write the area of $ A $ the rectangular region as a function of $ x $ What dimensions will produce a rectangle of maximum area?

a) See diagramb) $\frac{y}{2}, \pi y$c) $y=\frac{200-2 x}{\pi}$d) $50, \frac{100}{\pi}$

Algebra

Chapter 2

Polynomial and Rational Functions

Section 1

Quadratic Functions and Models

Quadratic Functions

Complex Numbers

Polynomials

Rational Functions

Missouri State University

Harvey Mudd College

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trying to illustrate the problem. Why is your ex is your land? And why is your with? So if you look at the inside the rectangle But why is representing the diameter in part A. This is your illustration part B radius. So next we will find the radius radius of the semicircle is going to be half of Why Since why is the diameter okay? Within that same question, we want to find the ah distance inside of the semicircle. So for a circle circumference is two pi r. But since we are looking at a semi circle, then we're going to say hi. You are because it's in my circle would be half of that. So the circumference for one, um, in my circle is going to be pi times 1/2. Why? So that gives us high over to times. Why heart see equation for a total distance traveled one lap. A one lap is the whole, too. We're going to say one lap around is really your perimeter. So the perimeter is equal to let's take the X plus the egg, so that's gonna be two X and then your distance around. This is a semi circle is a second semicircle, which technically gives you a circle. So too times high two times Hi. Over to why? So we took the answer from heart to include it for part C Till May told us that this is equal to 200. Okay, so let's clean this up. So we have two eggs two times this 1/2 is going to cancel out. So that's gonna be pi. Why equals 200? This equation has to be written in terms of why are in terms of X so we need to solve for why? So we're going to subtract two X. You have high times wise people 200 minus two x then to get why by it. So divide everything by pie so you're wise equals 200 minus two x divided by pi. So this represents part C. Okay, Heart d, That's the last part. Something that last we will area of just the rectangular region is X times Why so we have to in terms leads a function of X. So we need to find the dimensions X times your wise going to be taken from part C. So we're going to do 200 miners, two x about it by pi. Okay, so we can apply this to where we can use calculator, decimals, calculator to plug in the functions that we plug in the function without the area that we just plug in the right side of the function and then to find the dimensions that will give us the maximum area we're going to take. And look at the This is a pair of it. It turned tarsus to a parabola. So we're gonna take a look at the highest point, which you should for tech. So we're gonna take that X value That X equals 50 is where you will reach its maximum area. That is at the highest point on the graph. So we will take and use 50 for the X value. Love this into the Y value or to find your, uh, other dimension. So this gives you 5 50 times 200 minus 100 over pie 50 times, 100 over

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