Applications 3 - Optimization of Area and Volume PDF

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Warning: TT: undefined function: 32 Calculus and Vectors Optimization ProblemsName: ____________________ Date: ____________________Learning Goal Solve optimization problems involving area and volume.A very useful application of derivatives is optimization, that is finding the best (or worst) possibl...

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Calculus and Vectors Optimization Problems Learning Goal

Name: ____________________ Date: ____________________

Solve optimization problems involving area and volume.

A very useful application of derivatives is optimization, that is finding the best (or worst) possible outcome, subject to a set of restrictions. To find the optimal value, we find maximum and minimum values on a given interval. The First Derivative Test If 𝑓′󰇛󰇜 changes sign from negative to positive at   𝑐, then 𝑓󰇛󰇜 has a local minimum at this point.

If 𝑓′󰇛󰇜 changes sign from positive to negative at   𝑐, then 𝑓󰇛󰇜 has a local maximum at this point. If 𝑓′󰇛󰇜 does not change sign at   𝑐, then 𝑓󰇛󰇜 has no max/min here.

Example 1

If 2700 𝑚  of material is available to make a box with a square base and open top, find the largest possible volume of the box.

Example 2

Local farmers wish to sell apple cider in cylindrical containers. They need you to design a container that will hold 432 𝑚𝐿 of apple cider and minimize the cost of the metal used to manufacture the container. Determine the dimensions of this container.

Example 4

Find the dimensions of the largest right-cylinder that can be inscribed in a cone of radius 𝑟  3 𝑚 and height ℎ  6 𝑚.

Example 5

A rectangle is inscribed in a circle:       9. Find the dimensions of the rectangle with the maximum area.

Example 6

The roofline of the third floor of a house is in the shape of an equilateral triangle with each side measuring 16 feet. The home owners wish to install a rectangular glass door at one end of the roof. Determine the maximum area of the door that could be installed.

Practice 1.

A cone shaped paper drinking cup is to be made to hold 27 cm3 of water. Find the height and radius of the cup that will use the smallest amount of paper.

2.

Find the area of the largest rectangle that can be inscribed in a semicircle of radius 𝑟.

3.

Find the area of the largest rectangle that can be inscribed in the ellipse

 



     1.

4.

A trough is 16 feet long. Its cross sections are isosceles triangles with each of the two equal sides measuring 18 inches. Determine the dimensions (height and base) of the trough that maximizes its volume.

5.

Find the area of the largest isosceles triangle that can be inscribed in a circle of radius 6.

6.

Find the volume of the largest right circular cone that can be inscribed in a sphere of radius 𝑟....