1. 5-4 Day 1 modeling & optimization

2. How to Solve:
Understand the problem
Develop a model – draw a picture!
introduce a variable
write a function
Graph it (helps to decide what makes sense)
Find critical points & endpoints
Solve (plugging back in)
Interpret the solution find more answers
labels
sentence
Note: Most problems ask us to maximize or minimize a quantity
 derivatives!!! 

3. (max)
Ex 1) Find two numbers whose sum is 20 and whose product is as large as possible.
1st: x
2nd: 20 – x
Sum: 20
1st + 2nd = 20
Product:
f (x) = x(20 – x)
= 20x – x2
f ' (x) = 20 – 2x = 0
20 = 2x
10 = x
f " (x) = –2 < 0
1st: 10
2nd: 20 – 10 = 10
10 and 10
max!!

4. Ex 2) A rectangle is to be inscribed under one arch of the sine curve
Ex 2) A rectangle is to be inscribed under one arch of the sine curve. What is the largest area the rectangle can have, and what dimensions give that area?
(max area)
(x, sin x)
( – x, sin x) x 
w = sin (.710) = .652
l =  – 2(.710) = 1.722
Area = 1.123 by
Graph to find zeros
x = .710 because + –

5. Ex 3) An open-top box is to be made by cutting congruent squares of side length x from the corners of a 20- by 25-inch sheet of tin and bending up the sides. How large should the squares be to make the box hold as much as possible? What is the resulting maximum volume?
(volume = max) 20
V = l  w  h
V = (25 – 2x)(20 – 2x)(x)
V = 4x3 – 90x x
V' = 12x2 – 180x + 500
Quadratic Formula
x = , 3.681
domain [0, 10] x 25
Cut: in.
V = in3
25 – 2x x
20 – 2x

6. Ex 4) You have been asked to design a one-liter oil can shaped like a right circular cylinder. What dimensions will use the least material?
(min surface area)
r = cm
h = cm

7. homework
Pg # 23, 29, 47, 57
Pg # 2–32 (mult of 2+6n)