Optimization

Warm-up

Three quick questions from earlier lessons.

Discover

A farmer has $p$ m of fence for three sides of a rectangle against a wall. The graph shows the area $A(x) = x(p - 2x)$ for a width $x$ . Move $p$ .

The area A(x) = x(p − 2x) of a rectangle fenced on three sides with p meters. Each parabola peaks at x = p/4.

For $p = 40$ , where is the area largest?

At $x = 10$
At $x \approx 13.3$ , a square

Key idea

Turn the story into one function of one variable, then use the closed interval method.

Optimization: (1) name the quantity to optimize and the variables; (2) use the constraint to write it as a function of one variable; (3) find its domain; (4) find the critical numbers and compare with the endpoints; (5) answer the question asked, with units.

Worked example

Two positive numbers add to 20. Maximize their product.

P(x) = x(20 - x)

,

0 < x < 20

Constraint y = 20 − x.

x = 10

, a maximum (

P'' = -2

)

P'(x) = 20 − 2x.

Largest product

P(10) = 100

The question asks for the product.

Your turn, guided

Your turn: same problem with sum 30.

Largest product

225

x = 15 = y.

Practice ★

Two positive numbers add up to 35. What is the largest possible value of their product?
Two positive numbers add up to 20. What is the largest possible value of their product?

A farmer has 104 m of fence for three sides of a rectangle; a long wall forms the fourth side. Find the largest possible area, in m².
A farmer has 40 m of fence for three sides of a rectangle; a long wall forms the fourth side. Find the largest possible area, in m².

Watch out!

Practice ★★

Squares of side $x$ are cut from the corners of a 9-cm square sheet, and the sides are folded up into an open box. Which $x$ gives the largest volume?
Squares of side $x$ are cut from the corners of a 15-cm square sheet, and the sides are folded up into an open box. Which $x$ gives the largest volume?

Boss problem. Squares of side $x$ are cut from the corners of a 18-cm square sheet to fold an open box.

Exit ticket

Three questions to check you have got it.

A farmer has 104 m of fence for three sides of a rectangle; a long wall forms the fourth side. Find the largest possible area, in m².
A farmer has 40 m of fence for three sides of a rectangle; a long wall forms the fourth side. Find the largest possible area, in m².
A farmer has 60 m of fence for three sides of a rectangle; a long wall forms the fourth side. Find the largest possible area, in m².

Summary card

Optimization

Quantity and constraint.
One-variable function and its domain.
Critical numbers vs endpoints.
Answer the question: the value or the place, with units.

Watch out: not every critical number is the optimum.

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