Physical Fitness Facility An indoor running track is to be...

Name: Problem 72, Chapter 7: Precalculus with Limits
Uploaded: 2021-06-26T06:56:26-08:00
Duration: 5 min 16 s
Channel: Alex Roush
Description: Physical Fitness Facility An indoor running track is to be constructed with a space for exercise equipment inside the track (see figure). The track must be at least 125 meters long, and the exercise space must have an area of at least 500 square meters. (a) Find a system of inequalities describing the requirements of the facility. (b) Graph the system from part (a).

Physical Fitness Facility An indoor running track is to be constructed with a space for exercise equipment inside the track (see figure). The track must be at least 125 meters long, and the exercise space must have an area of at least 500 square meters. (a) Find a system of inequalities describing the requirements of the facility. (b) Graph the system from part (a).

Physical Fitness Facility An indoor running track
is to be constructed with a space for exercise equipment
inside the track (see figure). The track must be at least
125 meters long, and the exercise space must have an
area of at least 500 square meters.
(a) Find a system of inequalities describing the
requirements of the facility.
(b) Graph the system from part (a).

Key Concepts

Geometric Constraints

Geometric constraints refer to the conditions derived from the physical or spatial attributes of a problem, such as requirements for length, area, or volume. In facility design challenges, these constraints can include minimum lengths of paths or minimum areas for specific regions, and they serve as the boundaries within which the design must be feasible.

Systems of Inequalities

A system of inequalities is a collection of two or more inequalities that are considered simultaneously. In problems involving design constraints, such as minimum dimensions or area requirements, each constraint is represented by an inequality. The solution is the set of all values that satisfy every inequality in the system, ensuring that all practical conditions are met.

Graphing Feasible Regions

Graphing the feasible region involves plotting the inequalities on a coordinate plane to visually represent all possible solutions that satisfy the system of constraints. This graphical approach helps in understanding the interplay between different conditions, clearly delineating where all the requirements are met simultaneously.

Transcript

00:01 Okay, so we have this indoor track area that we want to put exercise equipment on and we have a couple of restraints.

00:07 So as you can see in the picture, it has this little rectangle and then this is a semicircle right here.

00:15 So these two semicircles will eventually make up one regular circle.

00:19 But the two constraints are that the total length, the total length of the track has to be at least 125 meters.

00:35 And the area, right, the area of this inside court right here is at least 500 meters.

00:55 Okay, so putting that to use, so this area has really nothing to do with the actual court, but we are considering the distance around.

01:06 Maybe you can think of this as like rails, and it's only thin enough to fit like a single person here, but there's really nothing over here.

01:15 Okay, so anyway, if the total length of the track has to be at least 125, we have to consider that we want this length and this length.

01:29 Okay, so this length should be pretty easy, right? there are just two parts to consider.

01:34 There's this longer side and then another longer side.

01:37 And the longer side is just represented by x.

01:40 So the length is just going to be two times x.

01:44 And that's of the rectangular portion.

01:45 Now, i said that these are both semicircles, and if you put them together, they make a larger circle.

01:51 So for the first part, you're going to have to run this little bit, this semicircle, and then you're going to take a break by running on the long rectangular portion, and then you're eventually going to run the rest of the circle.

02:04 So basically, you need to find the circumference of the circle.

02:07 And the circumference of the circle is just pi times diameter.

02:13 But in this case, the diameter is given by the side y.

02:19 So the circumference in this problem is pi times y.

02:22 So you run 2x plus pi y.

02:25 And that has to be at least 125.

02:30 And then if the area, this area right here has to be at least 500.

02:35 Well, the area of a rectangle is just base times height.

02:38 So that's going to be x, y is greater than or equal to 500.

02:43 Okay.

02:44 Now, graphing this is going to be a little tricky.

02:46 And i'm not going to do this exactly.