1. Modeling and Optimization
Section 4.4b

2. 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:

3. 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!!!

4. 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

5. 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…

6. 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

7. 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.

8. 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:

9. 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: