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# Week 11 Problem Set B.
## Reading.
Chapter 4.4 Concavity and curve sketching
Chapter 4.5 Applied optimization
Chapter 4.6 Newton's method
This problem set is on applied optimization (4.5).
Some strategy:
(1) Read the problem and draw a diagram
(2) Identify your objective function to maximize or minimize
(3) Label everything relevant with a variable
(4) Identify any constraints if any
(5) Once you have your objective function as a single variable function, then you can perform your usual method of finding extrema.
(6) Interpret and answer the original problem.
## Problems.
### Optimization.
1. A rectangle drawn on the xy-plane has its base on the $x$-axis and its upper two vertices on the parabola $y=12-x^{2}$. What is the largest area the rectangle can have, and what are its dimensions?
2. **The best fencing plan.** A rectangular plot of farmland will be bounded on one side by a river and on the other three sides by a single-strand electric fence. With 800 m of wire at your disposal, what is the largest area you can enclose, and what are its dimensions?
3. **Designing a poster.** You are designing a rectangular poster to contain 50 $\text{in}^{2}$ of printing with a 4-inch margin at the top and bottom and a 2-inch margin at each left- and right-side. What overall dimensions will minimize the amount of paper used?
4. **Quickest route.** Jane is 2 miles offshore in a boat and wishes to reach a costal village 6 miles down a straight shoreline from the point nearest the boat. She can row 2 miles per hour and can walk 5 miles per hour. Where should she land her boat to reach the village in the least amount of time?
5. Find a positive number for which the sum of **it** and **its reciprocal** is the smallest possible.
6. Find a positive number for which the sum of **its reciprocal** and **four times its square** is the smallest possible.
7. A piece of wire of 100 cm is to be cut into two pieces. One piece is bent into an equilateral triangle, and the other piece is bent into a square. What are the lengths of each piece so that the total area of the equilateral triangle and the square is the smallest?
8. Determine the dimension of the rectangle of largest area that can be inscribed in the right triangle shown here:![[1-teaching/smc-fall-2023-math-7/week-11/---files/Pasted image 20231110113929.png]]
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