Lesson 62 — Applied Optimization

With 100 m of fencing, what is the largest rectangular area that can be enclosed?

A rectangular pasture is divided in half by a fence parallel to the width. The total fencing (perimeter + divider) is 120 m. Maximize the area.

A cylindrical cup (bottom but no top) must have a volume of 500 cm³. What dimensions minimize the material used?

A rectangular box with a square base and no top must have a volume of 32 cm³. The base material costs R$ 2/cm² and the sides cost R$ 1/cm². Minimize the total cost.

The sum of two numbers is 20. Find the two numbers that maximize their product.

Find the positive number $$ such that the sum of $$ and its reciprocal multiplied by 4 is minimized.

Determine the point on the x-axis closest to the point $$.

Determine the point on the line $$ closest to the point $$.

From a 24 cm × 9 cm cardboard sheet, a square is cut from each corner and the flaps are folded up to form an open-top box. Determine the cut that maximizes the box's volume.

The demand function for a product is $$ (price in R$ per unit, $$ units sold). Maximize total revenue $$.

A company has revenue $$ and cost $$. Determine the production level that maximizes profit.

A box with a square base and a top must have a volume of 96 cm³. Minimize the total surface area.

In a constrained optimization problem, what is the correct role of the constraint equation?

To find the absolute maximum or minimum of $$ on $$, one must:

Find the dimensions of the cylinder with the largest volume that can be inscribed in a sphere of radius $$.

Find the points on the parabola $$ closest to the point $$.

A rectangular area of 300 m² is to be fenced. The east side (length $$) costs R$ 2/m, and the other sides cost R$ 3/m per meter. Minimize the total cost.

An object is launched vertically with an initial velocity of 20 m/s from a height of 3 m. Model: $$. Determine the maximum altitude.

A cylindrical can of volume 200 cm³ has top and bottom material costing R$ 10/cm² and side material costing R$ 6/cm². Determine the dimensions that minimize the cost.

A running track has the shape of a rectangle with semicircles on the two short ends (oval track). The total perimeter is 20 m. Determine the radius $$ that maximizes the enclosed area.

The sum of two numbers is 10. Find the two numbers that minimize the sum of their squares.

The sum of two non-negative numbers is 1. Maximize the product of the square of the first number and the second number.

Find the maximum area of a rectangle inscribed in a semicircle of radius 5.

Find the point on the curve $$ closest to the point $$.

An excursion charges R$ 80 per person for a group of 100. For each additional passenger, the fare for everyone drops R$ 0.50. How many passengers maximize revenue?

An orange grove with 25 trees per hectare produces 600 oranges per tree. For each additional tree planted, the yield per tree drops by 12 oranges. How many trees per hectare maximize the total orange production?

A "Norman window" is formed by a rectangle topped by a semicircle. The total perimeter is 10 m. Determine the radius of the semicircle that maximizes the window's area.

Use calculus to prove that, among all pairs of positive numbers with a fixed sum $$, the product is maximized when the two numbers are equal. (This proves the AM-GM inequality for two terms.)

Prove that the cylinder with the minimum surface area for a fixed volume $$ satisfies $$ (height equals diameter).

Find the maximum area of a rectangle inscribed in the ellipse $$, with sides parallel to the coordinate axes.