Glencoe Algebra 2 Student Edition C2014
Glencoe Algebra 2 Student Edition C2014
1st Edition
ISBN: 9780076639908
Author: McGraw-Hill Glencoe
Publisher: MCG
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Chapter 6.6, Problem 91SR
To determine

A rectangle is inscribed in an isosceles triangle as shown. Find the dimensions of the inscribed rectangle with maximum area.

Expert Solution & Answer
Check Mark

Answer to Problem 91SR

The dimensions of the rectangle when its area is maximum are 5 inches and 4 inches.

Explanation of Solution

Given:

The diagram is :

  Glencoe Algebra 2 Student Edition C2014, Chapter 6.6, Problem 91SR , additional homework tip  1

Concept Used:

  • To maximize a function f ( x ), take its derivative wrt x and equate it to 0:

  ddx[f(x)]=0 and find the value of x .

If d2dx2[f(x)]<0 , then the value is maximum.

Calculation:

In the given triangle, let

2b = length of the base of the triangle = 10 in.

h = height of the triangle = 8 in.

In the given rectangle , let

2x = length of the base adjacent to the base of the triangle

y = width of the rectangle

  Glencoe Algebra 2 Student Edition C2014, Chapter 6.6, Problem 91SR , additional homework tip  2

The triangles ΔABD and ΔFBE are similar .

So, ADFE=BDBEhy=bbxh(bx)=by..................[multiply each side by y(b-x)]y=h(bx)b..................[divide each side by b]y=hhxb

The area of the rectangle is:

  length×width=2xy=2x(hhxb)......................[y=hhxb]=2xh2hx2ba(x)=2xh2hx2b

Now, we need to maximize the area of the rectangle . So, we will derivative the function that represents the area of the rectangle wrt x and equate it to 0:

  ddx[a(x)]=0ddx[2xh2hx2b]=02h4hxb=0..................[since d2dx2(a(x))=4hb<0,the solution is a point of maxima]2hb4hx=0................[multiply both sides by b]2hb=4hx.....................[add 4hx each side]4hx2h=2hb2h...................[divide each side by 2h]2x=bx=b2...........................[divide each side by 2]andy=hhxby=hhb(b2)............[x=b2]y=hh2y=h2

Hence,

The length of the rectangle = 2x=2(b2)=b=102=5 inches.

The width of the rectangle = y=h2=82=4 inches.

So, the dimensions of the rectangle when its area is maximum are 5 inches and 4 inches.

Chapter 6 Solutions

Glencoe Algebra 2 Student Edition C2014

Ch. 6.1 - Prob. 4CYUCh. 6.1 - Prob. 5CYUCh. 6.1 - Prob. 6CYUCh. 6.1 - Prob. 7CYUCh. 6.1 - Prob. 8PPSCh. 6.1 - Prob. 9PPSCh. 6.1 - Prob. 10PPSCh. 6.1 - Prob. 11PPSCh. 6.1 - Prob. 12PPSCh. 6.1 - Prob. 13PPSCh. 6.1 - Prob. 14PPSCh. 6.1 - Prob. 15PPSCh. 6.1 - Prob. 16PPSCh. 6.1 - Prob. 17PPSCh. 6.1 - Prob. 18PPSCh. 6.1 - Prob. 19PPSCh. 6.1 - Prob. 20PPSCh. 6.1 - Prob. 21PPSCh. 6.1 - Prob. 22PPSCh. 6.1 - Prob. 23PPSCh. 6.1 - Prob. 24PPSCh. 6.1 - Prob. 25PPSCh. 6.1 - Prob. 26PPSCh. 6.1 - Prob. 27PPSCh. 6.1 - Prob. 28PPSCh. 6.1 - Prob. 29PPSCh. 6.1 - Prob. 30PPSCh. 6.1 - Prob. 31PPSCh. 6.1 - Prob. 32PPSCh. 6.1 - Prob. 33PPSCh. 6.1 - Prob. 34PPSCh. 6.1 - Prob. 35PPSCh. 6.1 - Prob. 36PPSCh. 6.1 - Prob. 37PPSCh. 6.1 - Prob. 38PPSCh. 6.1 - Prob. 39PPSCh. 6.1 - Prob. 40PPSCh. 6.1 - Prob. 41PPSCh. 6.1 - Prob. 42PPSCh. 6.1 - Prob. 43PPSCh. 6.1 - Prob. 44PPSCh. 6.1 - Prob. 45PPSCh. 6.1 - Prob. 46PPSCh. 6.1 - Prob. 47PPSCh. 6.1 - Prob. 48PPSCh. 6.1 - Prob. 49PPSCh. 6.1 - Prob. 50PPSCh. 6.1 - Prob. 51PPSCh. 6.1 - 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Prob. 3ECh. 6.4 - Prob. 4ECh. 6.4 - Prob. 5ECh. 6.4 - Prob. 6ECh. 6.4 - Prob. 7ECh. 6.4 - Prob. 8ECh. 6.4 - Prob. 1MCQCh. 6.4 - Prob. 2MCQCh. 6.4 - Prob. 3MCQCh. 6.4 - Prob. 4MCQCh. 6.4 - Prob. 5MCQCh. 6.4 - Prob. 6MCQCh. 6.4 - Prob. 7MCQCh. 6.4 - Prob. 8MCQCh. 6.4 - Prob. 9MCQCh. 6.4 - Prob. 10MCQCh. 6.4 - Prob. 11MCQCh. 6.4 - Prob. 12MCQCh. 6.4 - Prob. 13MCQCh. 6.4 - Prob. 14MCQCh. 6.4 - Prob. 15MCQCh. 6.4 - Prob. 16MCQCh. 6.4 - Prob. 17MCQCh. 6.4 - Prob. 18MCQCh. 6.4 - Prob. 19MCQCh. 6.4 - Prob. 20MCQCh. 6.4 - Prob. 21MCQCh. 6.4 - Prob. 22MCQCh. 6.4 - Prob. 23MCQCh. 6.4 - Prob. 24MCQCh. 6.4 - Prob. 25MCQCh. 6.4 - Prob. 26MCQCh. 6.4 - Prob. 27MCQCh. 6.4 - Prob. 28MCQCh. 6.4 - Prob. 29MCQCh. 6.4 - Prob. 30MCQCh. 6.4 - Prob. 31MCQCh. 6.5 - Prob. 1AGPCh. 6.5 - Prob. 1BGPCh. 6.5 - Prob. 2AGPCh. 6.5 - Prob. 2BGPCh. 6.5 - Prob. 3AGPCh. 6.5 - Prob. 3BGPCh. 6.5 - Prob. 4AGPCh. 6.5 - Prob. 4BGPCh. 6.5 - Prob. 5AGPCh. 6.5 - Prob. 5BGPCh. 6.5 - Prob. 6GPCh. 6.5 - Prob. 1CYUCh. 6.5 - Prob. 2CYUCh. 6.5 - 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Prob. 50PPSCh. 6.6 - Prob. 51PPSCh. 6.6 - Prob. 52PPSCh. 6.6 - Prob. 53PPSCh. 6.6 - Prob. 54PPSCh. 6.6 - Prob. 55PPSCh. 6.6 - Prob. 56PPSCh. 6.6 - Prob. 57PPSCh. 6.6 - Prob. 58PPSCh. 6.6 - Prob. 59PPSCh. 6.6 - Prob. 60PPSCh. 6.6 - Prob. 61PPSCh. 6.6 - Prob. 62PPSCh. 6.6 - Prob. 63PPSCh. 6.6 - Prob. 64PPSCh. 6.6 - Prob. 65PPSCh. 6.6 - Prob. 66HPCh. 6.6 - Prob. 67HPCh. 6.6 - Prob. 68HPCh. 6.6 - Prob. 69HPCh. 6.6 - Prob. 70HPCh. 6.6 - Prob. 71STPCh. 6.6 - Prob. 72STPCh. 6.6 - Prob. 73STPCh. 6.6 - Prob. 74STPCh. 6.6 - Prob. 75SRCh. 6.6 - Prob. 76SRCh. 6.6 - Prob. 77SRCh. 6.6 - Prob. 78SRCh. 6.6 - Prob. 79SRCh. 6.6 - Prob. 80SRCh. 6.6 - Prob. 81SRCh. 6.6 - Prob. 82SRCh. 6.6 - Prob. 83SRCh. 6.6 - Prob. 84SRCh. 6.6 - Prob. 85SRCh. 6.6 - Prob. 86SRCh. 6.6 - Prob. 87SRCh. 6.6 - Prob. 88SRCh. 6.6 - Prob. 89SRCh. 6.6 - Prob. 90SRCh. 6.6 - Prob. 91SRCh. 6.6 - Prob. 92SCh. 6.6 - Prob. 93SCh. 6.6 - Prob. 94SCh. 6.6 - Prob. 95SCh. 6.6 - Prob. 96SCh. 6.6 - Prob. 97SCh. 6.7 - Prob. 1AGPCh. 6.7 - Prob. 1BGPCh. 6.7 - Prob. 2AGPCh. 6.7 - Prob. 2BGPCh. 6.7 - Prob. 3GPCh. 6.7 - Prob. 4AGPCh. 6.7 - Prob. 4BGPCh. 6.7 - Prob. 1CYUCh. 6.7 - Prob. 2CYUCh. 6.7 - Prob. 3CYUCh. 6.7 - Prob. 4CYUCh. 6.7 - Prob. 5CYUCh. 6.7 - Prob. 6CYUCh. 6.7 - Prob. 7CYUCh. 6.7 - Prob. 8CYUCh. 6.7 - Prob. 9CYUCh. 6.7 - Prob. 10CYUCh. 6.7 - Prob. 11CYUCh. 6.7 - Prob. 12CYUCh. 6.7 - Prob. 13CYUCh. 6.7 - Prob. 14CYUCh. 6.7 - Prob. 15CYUCh. 6.7 - Prob. 16CYUCh. 6.7 - Prob. 17CYUCh. 6.7 - Prob. 18CYUCh. 6.7 - Prob. 19CYUCh. 6.7 - Prob. 20CYUCh. 6.7 - Prob. 21CYUCh. 6.7 - Prob. 22CYUCh. 6.7 - Prob. 23PPSCh. 6.7 - Prob. 24PPSCh. 6.7 - Prob. 25PPSCh. 6.7 - Prob. 26PPSCh. 6.7 - Prob. 27PPSCh. 6.7 - Prob. 28PPSCh. 6.7 - Prob. 29PPSCh. 6.7 - Prob. 30PPSCh. 6.7 - Prob. 31PPSCh. 6.7 - Prob. 32PPSCh. 6.7 - Prob. 33PPSCh. 6.7 - Prob. 34PPSCh. 6.7 - Prob. 35PPSCh. 6.7 - Prob. 36PPSCh. 6.7 - Prob. 37PPSCh. 6.7 - Prob. 38PPSCh. 6.7 - Prob. 39PPSCh. 6.7 - Prob. 40PPSCh. 6.7 - Prob. 41PPSCh. 6.7 - Prob. 42PPSCh. 6.7 - Prob. 43PPSCh. 6.7 - Prob. 44PPSCh. 6.7 - Prob. 45PPSCh. 6.7 - Prob. 46PPSCh. 6.7 - Prob. 47PPSCh. 6.7 - Prob. 48PPSCh. 6.7 - Prob. 49PPSCh. 6.7 - Prob. 50PPSCh. 6.7 - Prob. 51PPSCh. 6.7 - Prob. 52PPSCh. 6.7 - Prob. 53PPSCh. 6.7 - Prob. 54PPSCh. 6.7 - Prob. 55PPSCh. 6.7 - Prob. 56PPSCh. 6.7 - Prob. 57PPSCh. 6.7 - Prob. 58PPSCh. 6.7 - Prob. 59PPSCh. 6.7 - Prob. 60PPSCh. 6.7 - Prob. 61PPSCh. 6.7 - Prob. 62PPSCh. 6.7 - Prob. 63PPSCh. 6.7 - Prob. 64PPSCh. 6.7 - Prob. 65PPSCh. 6.7 - Prob. 66PPSCh. 6.7 - Prob. 67HPCh. 6.7 - Prob. 68HPCh. 6.7 - Prob. 69HPCh. 6.7 - Prob. 70HPCh. 6.7 - Prob. 71HPCh. 6.7 - Prob. 72HPCh. 6.7 - Prob. 73HPCh. 6.7 - Prob. 74HPCh. 6.7 - Prob. 75HPCh. 6.7 - Prob. 76STPCh. 6.7 - Prob. 77STPCh. 6.7 - Prob. 78STPCh. 6.7 - Prob. 79STPCh. 6.7 - Prob. 80SRCh. 6.7 - Prob. 81SRCh. 6.7 - Prob. 82SRCh. 6.7 - Prob. 83SRCh. 6.7 - Prob. 84SRCh. 6.7 - Prob. 85SRCh. 6.7 - Prob. 86SRCh. 6.7 - Prob. 87SRCh. 6.7 - Prob. 88SRCh. 6.7 - Prob. 89SRCh. 6.7 - Prob. 90SRCh. 6.7 - Prob. 91SCh. 6.7 - Prob. 92SCh. 6.7 - Prob. 93SCh. 6.7 - Prob. 94SCh. 6.7 - Prob. 95SCh. 6.7 - Prob. 96SCh. 6.7 - Prob. 97SCh. 6.7 - Prob. 98SCh. 6.7 - Prob. 1ECh. 6.7 - Prob. 2ECh. 6.7 - Prob. 3ECh. 6.7 - Prob. 4ECh. 6.7 - Prob. 5ECh. 6.7 - Prob. 6ECh. 6.7 - Prob. 7ECh. 6.7 - Prob. 8ECh. 6.7 - Prob. 9ECh. 6.7 - Prob. 10ECh. 6 - Prob. 1QCCh. 6 - Prob. 2QCCh. 6 - Prob. 3QCCh. 6 - Prob. 4QCCh. 6 - Prob. 5QCCh. 6 - Prob. 6QCCh. 6 - Prob. 7QCCh. 6 - Prob. 8QCCh. 6 - Prob. 1SGRCh. 6 - Prob. 2SGRCh. 6 - Prob. 3SGRCh. 6 - Prob. 4SGRCh. 6 - Prob. 5SGRCh. 6 - Prob. 6SGRCh. 6 - Prob. 7SGRCh. 6 - Prob. 8SGRCh. 6 - Prob. 9SGRCh. 6 - Prob. 10SGRCh. 6 - Prob. 11SGRCh. 6 - Prob. 12SGRCh. 6 - Prob. 13SGRCh. 6 - Prob. 14SGRCh. 6 - Prob. 15SGRCh. 6 - Prob. 16SGRCh. 6 - Prob. 17SGRCh. 6 - Prob. 18SGRCh. 6 - Prob. 19SGRCh. 6 - Prob. 20SGRCh. 6 - Prob. 21SGRCh. 6 - Prob. 22SGRCh. 6 - Prob. 23SGRCh. 6 - Prob. 24SGRCh. 6 - Prob. 25SGRCh. 6 - Prob. 26SGRCh. 6 - Prob. 27SGRCh. 6 - Prob. 28SGRCh. 6 - Prob. 29SGRCh. 6 - Prob. 30SGRCh. 6 - Prob. 31SGRCh. 6 - Prob. 32SGRCh. 6 - Prob. 33SGRCh. 6 - Prob. 34SGRCh. 6 - Prob. 35SGRCh. 6 - Prob. 36SGRCh. 6 - Prob. 37SGRCh. 6 - Prob. 38SGRCh. 6 - Prob. 39SGRCh. 6 - Prob. 40SGRCh. 6 - Prob. 41SGRCh. 6 - Prob. 42SGRCh. 6 - Prob. 43SGRCh. 6 - Prob. 44SGRCh. 6 - Prob. 45SGRCh. 6 - Prob. 46SGRCh. 6 - Prob. 47SGRCh. 6 - Prob. 48SGRCh. 6 - Prob. 49SGRCh. 6 - Prob. 50SGRCh. 6 - Prob. 51SGRCh. 6 - Prob. 52SGRCh. 6 - Prob. 53SGRCh. 6 - Prob. 54SGRCh. 6 - Prob. 55SGRCh. 6 - Prob. 56SGRCh. 6 - Prob. 57SGRCh. 6 - Prob. 58SGRCh. 6 - Prob. 59SGRCh. 6 - Prob. 60SGRCh. 6 - Prob. 61SGRCh. 6 - Prob. 62SGRCh. 6 - Prob. 63SGRCh. 6 - Prob. 64SGRCh. 6 - Prob. 65SGRCh. 6 - Prob. 66SGRCh. 6 - Prob. 67SGRCh. 6 - Prob. 68SGRCh. 6 - Prob. 69SGRCh. 6 - Prob. 70SGRCh. 6 - Prob. 71SGRCh. 6 - Prob. 72SGRCh. 6 - Prob. 73SGRCh. 6 - Prob. 74SGRCh. 6 - Prob. 75SGRCh. 6 - Prob. 76SGRCh. 6 - Prob. 77SGRCh. 6 - Prob. 78SGRCh. 6 - Prob. 79SGRCh. 6 - Prob. 80SGRCh. 6 - Prob. 81SGRCh. 6 - Prob. 1PTCh. 6 - Prob. 2PTCh. 6 - Prob. 3PTCh. 6 - Prob. 4PTCh. 6 - Prob. 5PTCh. 6 - Prob. 6PTCh. 6 - Prob. 7PTCh. 6 - Prob. 8PTCh. 6 - Prob. 9PTCh. 6 - Prob. 10PTCh. 6 - Prob. 11PTCh. 6 - Prob. 12PTCh. 6 - Prob. 13PTCh. 6 - Prob. 14PTCh. 6 - Prob. 15PTCh. 6 - Prob. 16PTCh. 6 - Prob. 17PTCh. 6 - Prob. 18PTCh. 6 - Prob. 19PTCh. 6 - Prob. 20PTCh. 6 - Prob. 21PTCh. 6 - Prob. 22PTCh. 6 - Prob. 23PTCh. 6 - Prob. 24PTCh. 6 - Prob. 25PTCh. 6 - Prob. 26PTCh. 6 - Prob. 27PTCh. 6 - Prob. 28PTCh. 6 - Prob. 29PTCh. 6 - Prob. 30PTCh. 6 - Prob. 31PTCh. 6 - Prob. 32PTCh. 6 - Prob. 33PTCh. 6 - Prob. 34PTCh. 6 - Prob. 35PTCh. 6 - Prob. 36PTCh. 6 - Prob. 37PTCh. 6 - Prob. 38PTCh. 6 - Prob. 39PTCh. 6 - Prob. 40PTCh. 6 - Prob. 1ECh. 6 - Prob. 2ECh. 6 - Prob. 3ECh. 6 - Prob. 1STPCh. 6 - Prob. 2STPCh. 6 - Prob. 3STPCh. 6 - Prob. 4STPCh. 6 - Prob. 5STPCh. 6 - Prob. 6STPCh. 6 - Prob. 7STPCh. 6 - Prob. 8STPCh. 6 - Prob. 9STPCh. 6 - Prob. 10STPCh. 6 - Prob. 11STPCh. 6 - Prob. 12STPCh. 6 - Prob. 13STPCh. 6 - Prob. 14STPCh. 6 - Prob. 15STP
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