4.2: Applied Optimization Problems
- Page ID
- 144281
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- You have \(400 \text{m}\) of fencing to construct a pen for cattle. What should the dimensions of the pen be to maximize the area of the enclosure? What is the maximum possible area of the enclosure?
- What's the smallest possible amount of fencing needed to enclose a rectangular area of \(225 \text{ft}^2\)?
- You have \(100 \text{ft}\) of fencing to make a rectangular garden alongside the wall of your house. The wall of your house will bound one side of the garden. What should the dimensions of garden be to maximize its area? What is the largest possible area of the garden?
- A box with a square base and no top is to hold a volume of \(100\ \text{cm}^3\). Find the dimensions of the box that requires the least amount of material to construct the five sides of the box.
- If a cylinder has a surface area of \(50 \text{ft}^2\), including the top and bottom, what is the maximum possible volume of the cylinder?
- You are constructing an open-top box with a square bottom for your cat to sleep in. The plush material you want to use for the bottom of the box costs \(\$5\) per square foot. The material you want to use for the sides of the box cost \(\$ 2\) per square foot. The volume of the box must be \(4 \text{ft}^3\). Find the dimensions of the box that minimize the cost to construct it.
- Find the dimensions of the lightest cylindrical can that contains \(250 \text{cm}^3\) if the top and bottom of the can are made from a material twice as heavy (per unit area) as the material used for the side of the can.
- You run a limousine rental company. The gas mileage a limousine gets driving at speed \(v\) is \(m(v) = \dfrac{120 - 2v}{5} \text{mi}/\text{gal}\). You pay a chauffeur \(\$ 15\) per hour to drive the limousine, and gas costs \(\$3.50\) per gallon. Find your cost per mile at speed \(v\). What is the cheapest driving speed?