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Modeling and Optimization. Section 4.4b. Do Now: #2 on p.214. What is the largest possible area for a right triangle whose hypotenuse is 5 cm long, and what are its dimensions?. 5. Domain:. Derivative:. Do Now: #2 on p.214. What is the largest possible area for a right triangle whose

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Presentation Transcript
slide2

Do Now: #2 on p.214

What is the largest possible area for a right triangle whose

hypotenuse is 5 cm long, and what are its dimensions?

5

Domain:

Derivative:

slide3

Do Now: #2 on p.214

What is the largest possible area for a right triangle whose

hypotenuse is 5 cm long, and what are its dimensions?

Critical Point:

for

for

This critical point corresponds to a maximum area!!!

slide4

Do Now: #2 on p.214

What is the largest possible area for a right triangle whose

hypotenuse is 5 cm long, and what are its dimensions?

Solve for y:

Solve for A:

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

are by .

slide5

More Practice Problems: #18 on p.215

A piece of cardboard measures 10- by 15-in. Two equal squares

are removed from the corners of a 10-in. side as shown in the

figure. Two equal rectangles are removed from the other corners

so that the tabs can be folded to form a rectangular box with lid.

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

The base measures in. by in…

slide6

More Practice Problems: #18 on p.215

A piece of cardboard measures 10- by 15-in. Two equal squares

are removed from the corners of a 10-in. side as shown in the

figure. Two equal rectangles are removed from the other corners

so that the tabs can be folded to form a rectangular box with lid.

(b) Find the domain and graph of .

Graph in by

(c) Find the maximum volume graphically.

The maximum volume is approximately

when

slide7

More Practice Problems: #18 on p.215

A piece of cardboard measures 10- by 15-in. Two equal squares

are removed from the corners of a 10-in. side as shown in the

figure. Two equal rectangles are removed from the other corners

so that the tabs can be folded to form a rectangular box with lid.

(d) Confirm this answer analytically.

when

at our critical point, meaning that this

point corresponds to a maximum volume.

slide8

More Practice Problems: #36 on p.217

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

(Hint: If you minimize the square of the distance, you can avoid

square roots.)

The square of the distance:

Domain:

slide9

More Practice Problems: #36 on p.217

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

(Hint: If you minimize the square of the distance, you can avoid

square roots.)

Domain:

Minimize analytically:

CP:

Since changes sign from negative to positive at ,

the critical point corresponds to a minimum distance.

Minimum distance: