Modeling and Optimization

Section 4.4b

What is the largest possible area for a right triangle whosehypotenuse is 5 cm long, and what are its dimensions?

5

2 2 25x y 225y x 1

2A xy 21

252x x

Domain: 0,5

2

2

1 1 12 25

2 22 25

dAx x x

dx x

Derivative:

2 2

2 2

25

2 25 2 25

x x

x x

2

2

25 2

2 25

x

x

Do Now: #2 on p.214

x

y

What is the largest possible area for a right triangle whosehypotenuse is 5 cm long, and what are its dimensions?

Do Now: #2 on p.214

225 2 0x Critical Point:2

2

25 2

2 25

dA x

dx x

5

2x

0dA

dx for

50

2x 0

dA

dx for

55

2x

This critical point corresponds to a maximum area!!!

What is the largest possible area for a right triangle whosehypotenuse is 5 cm long, and what are its dimensions?

Do Now: #2 on p.214

Solve for y:2

525

2y

2525

2

25

2

5

2

Solve for A: 1 1 5 5

2 2 2 2A xy

25

4

The largest possible area is , and the dimensions (legs)225

cm4

are by .5cm2

5cm2

A piece of cardboard measures 10- by 15-in. Two equal squaresare removed from the corners of a 10-in. side as shown in thefigure. Two equal rectangles are removed from the other cornersso that the tabs can be folded to form a rectangular box with lid.

More Practice Problems: #18 on p.215

(a) Write a formula for the volume of the box. V x

The base measures in. by in…10 2x15 2

2

x

10 2 15 2

2

x x xV x

3 22 25 75x x x

A piece of cardboard measures 10- by 15-in. Two equal squaresare removed from the corners of a 10-in. side as shown in thefigure. Two equal rectangles are removed from the other cornersso that the tabs can be folded to form a rectangular box with lid.

More Practice Problems: #18 on p.215

(b) Find the domain and graph of . V x

: 0,5D Graph in by 0,5 V x 20,80

(c) Find the maximum volume graphically.

The maximum volume is approximately366.019in

when 1.962inx

A piece of cardboard measures 10- by 15-in. Two equal squaresare removed from the corners of a 10-in. side as shown in thefigure. Two equal rectangles are removed from the other cornersso that the tabs can be folded to form a rectangular box with lid.

More Practice Problems: #18 on p.215

(d) Confirm this answer analytically.

3 22 25 75V x x x x

26 50 75V x x x when 0V x 1.962,6.371x

12 50V x x

0V x at our critical point, meaning that this

point corresponds to a maximum volume.

How close does the curve come to the point (3/2, 0)?

More Practice Problems: #36 on p.217

(Hint: If you minimize the square of the distance, you can avoidsquare roots.)

y x

2

230

2D x x x

The square of the distance:

2 3 3 9

2 2 4x x x x 2 9

24

x x

0,Domain:

How close does the curve come to the point (3/2, 0)?

More Practice Problems: #36 on p.217

(Hint: If you minimize the square of the distance, you can avoidsquare roots.)

y x

2 92

4D x x x 0,Domain:

2 2D x x Minimize analytically: 0 1x CP:

D xSince changes sign from negative to positive at ,the critical point corresponds to a minimum distance.

1x

1DMinimum distance: 2 91 2 1

4

5

2