A new running track is to be constructed in the shape of a...

Name: Problem 27, Chapter 1: Precalculus
Uploaded: 2020-09-11T02:32:59-08:00
Duration: 4 min 39 s
Channel: Camille Riebel
Description: A new running track is to be constructed in the shape of a rectangle with semicircles at each end. The track is to be 440 yards long. Express the area of the region enclosed by the track, A, as a function of its radius, r.

A new running track is to be constructed in the shape of a rectangle with semicircles at each end. The track is to be 440 yards long. Express the area of the region enclosed by the track, A, as a function of its radius, r.

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Key Concepts

Formulating Functions from Geometric Relationships

After determining the relationships between dimensions from the perimeter constraint, the next step is to substitute these into the area formulas for the individual shapes. This process results in an expression for the total area solely as a function of the radius, illustrating how algebra and geometry work together to model real-world shapes.

Composite Figures

A composite figure refers to a shape that can be broken down into simpler shapes whose areas we can calculate individually. In this context, the track consists of a rectangular portion and two semicircular sections. Understanding how to add the areas of these simpler shapes to get the total area is key.

Perimeter Constraint and Variable Substitution

A perimeter (or boundary length) constraint relates different parts of a composite figure via a common measurement. Here, the total length of the track is given, and the portions of the track corresponding to the semicircles are related to the radius. This relationship is used to express one dimension (e.g., the length of the rectangle) in terms of the radius.

Transcript

00:02 So in this problem, we're given the shape of a track that is rectangular with two semicircles on the ends.

00:09 So we're going to start drawing our shape.

00:16 So each semicircle has a radius, and we're going to denote that as r, and that is on both sides.

00:27 And r is just half of the width of the track.

00:31 The distance between each of the semicircles would be the length of a rectangle, and we're going to denote that as x.

00:41 First thing that we're going to want to do is calculate the perimeter.

00:48 We're going to denote perimeter as p.

00:50 So we're going to go around our shape and calculate the perimeter.

00:55 So we're going to start at the top and that straight length right here is x.

01:01 So then we go to the semicircle and the perimeter of a semicircle is just pi r.

01:09 The perimeter of the entire circle is 2 pi r.

01:10 The perimeter of the entire circle is 2 pi r.

01:12 So a semicircle is just half.

01:15 So we add pi r.

01:18 And we keep going around our shape.

01:20 And we have another straight length of our rectangle, which is x.

01:25 So we add another x.

01:27 And then on the other side, the perimeter of this semicircle is the same as the original.

01:33 We're going to denote that as pi r.

01:39 So now we can simplify this equation, a group...

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