Chapter 3: Q. 17 (page 166)

Constructing a Stadium A track and field playing area is in the shape of a rectangle with semicircles at each end. See the figure. The inside perimeter of the track is to be $1500$ meters. What should the dimensions of the rectangle be so that the area of the rectangle is a maximum?

Short Answer

Expert verified

The area of a rectangle is maximized at $L = 375 m, w = 238.7 m .$

Step by step solution

Step 1. Given information

It is given that the inside perimeter of the truck is to be $1500$ meters. We need to determine the dimensions of the rectangle be so that the area of rectangle is maximum.

Step 2. Simplify

The figure, $r$ represents the radius, $l$ represents the length and $w$ represents the width.

The perimeter of the figure,

$P = 2 l + 2 (\frac{2 πr}{2}) .$

$= 2 l + 2 πr \dots (1) .$

We know, $w = 2 r .$

$P = 2 l + πw .$

The area of the rectangle is given by the equation.

$A = l w \dots (2) .$

We know, $P = 1500$ in (1).

$1500 = 2 l + πw .$

$2 l = 1500 - πw .$

$l = 750 - \frac{πw}{2} .$

Substitute $l$ in (2).

$A = (750 - \frac{πw}{2}) w .$

$= 750 w - \frac{{πw}^{2}}{2} .$

$= \frac{- {πw}^{2}}{2} + 750 w .$

The $w$ coordinate of the vertex of an equation in the form $a w^{2} - b w + c .$

$w = - \frac{b}{2 a} .$

$w = - \frac{750}{2 (- \frac{π}{2})} .$

$= \approx 238.7 m .$

Substitute $w$ in equaion 3.

$l = 750 - \frac{π (238.7)}{2} .$

$= 375 m .$

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Constructing a Stadium A track and field playing area is in the shape of a rectangle with semicircles at each end. See the figure. The inside perimeter of the track is to be $1500$ meters. What should the dimensions of the rectangle be so that the area of the rectangle is a maximum?

Short Answer

Step by step solution

Step 1. Given information

Step 2. Simplify

One App. One Place for Learning.

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Constructing a Stadium A track and field playing area is in the shape of a rectangle with semicircles at each end. See the figure. The inside perimeter of the track is to be 1500meters. What should the dimensions of the rectangle be so that the area of the rectangle is a maximum?

Short Answer

Step by step solution

Step 1. Given information

Step 2. Simplify

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Discrete Mathematics

Geometry

Decision Maths

Applied Mathematics

Probability and Statistics

Pure Maths

Study anywhere. Anytime. Across all devices.

Constructing a Stadium A track and field playing area is in the shape of a rectangle with semicircles at each end. See the figure. The inside perimeter of the track is to be $1500$ meters. What should the dimensions of the rectangle be so that the area of the rectangle is a maximum?