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# 6.E: Applications of the Derivative (Exercises)

These are homework exercises to accompany David Guichard's "General Calculus" Textmap. Complementary General calculus exercises can be found for other Textmaps and can be accessed here

## 6.1: Optimization

Ex 6.1.1   Let $$f(x) = \cases{1 + 4 x -x^2 & for x\leq3 \cr (x+5)/2 &for \(x>3$$ \cr}\).

Find the maximum value and minimum values of $$f(x)$$ for $$x$$ in $$[0,4]$$. Graph $$f(x)$$ to check your answers. (answer)

Ex 6.1.2   Find the dimensions of the rectangle of largest area having fixed perimeter 100. (answer)

Ex 6.1.3   Find the dimensions of the rectangle of largest area having fixed perimeter P. (answer)

Ex 6.1.4   A box with square base and no top is to hold a volume 100. Find the dimensions of the box that requires the least material for the five sides. Also find the ratio of height to side of the base. (answer)

Ex 6.1.5   A box with square base is to hold a volume 200. The bottom and top are formed by folding in flaps from all four sides, so that the bottom and top consist of two layers of cardboard. Find the dimensions of the box that requires the least material. Also find the ratio of height to side of the base. (answer)

Ex 6.1.6   A box with square base and no top is to hold a volume $$V$$. Find (in terms of $$V$$) the dimensions of the box that requires the least material for the five sides. Also find the ratio of height to side of the base. (This ratio will not involve $$V$$.) (answer)

Ex 6.1.7   You have 100 feet of fence to make a rectangular play area alongside the wall of your house. The wall of the house bounds one side. What is the largest size possible (in square feet) for the play area? (answer)

Ex 6.1.8   You have $$l$$ feet of fence to make a rectangular play area alongside the wall of your house. The wall of the house bounds one side. What is the largest size possible (in square feet) for the play area? (answer)

Ex 6.1.9   Marketing tells you that if you set the price of an item at $10 then you will be unable to sell it, but that you can sell 500 items for each dollar below$10 that you set the price. Suppose your fixed costs total $3000, and your marginal cost is$2 per item. What is the most profit you can make?(answer)

Ex 6.1.10   Find the area of the largest rectangle that fits inside a semicircle of radius $$10$$ (one side of the rectangle is along the diameter of the semicircle). (answer)

Ex 6.1.11   Find the area of the largest rectangle that fits inside a semicircle of radius $$r$$ (one side of the rectangle is along the diameter of the semicircle). (answer)

Ex 6.1.12   For a cylinder with surface area 50, including the top and the bottom, find the ratio of height to base radius that maximizes the volume. (answer)

Ex 6.1.13   For a cylinder with given surface area $$S$$, including the top and the bottom, find the ratio of height to base radius that maximizes the volume. (answer)

Ex 6.1.14   You want to make cylindrical containers to hold 1 liter using the least amount of construction material. The side is made from a rectangular piece of material, and this can be done with no material wasted. However, the top and bottom are cut from squares of side $$2r$$, so that $$2(2r)^2=8r^2$$ of material is needed (rather than $$2\pi r^2$$, which is the total area of the top and bottom). Find the dimensions of the container using the least amount of material, and also find the ratio of height to radius for this container. (answer)

Ex 6.1.15   You want to make cylindrical containers of a given volume $$V$$ using the least amount of construction material. The side is made from a rectangular piece of material, and this can be done with no material wasted. However, the top and bottom are cut from squares of side $$2r$$, so that $$2(2r)^2=8r^2$$ of material is needed (rather than $$2\pi r^2$$, which is the total area of the top and bottom). Find the optimal ratio of height to radius. (answer)

Ex 6.1.16   Given a right circular cone, you put an upside-down cone inside it so that its vertex is at the center of the base of the larger cone and its base is parallel to the base of the larger cone. If you choose the upside-down cone to have the largest possible volume, what fraction of the volume of the larger cone does it occupy? (Let $$H$$ and $$R$$ be the height and base radius of the larger cone, and let $$h$$ and $$r$$ be the height and base radius of the smaller cone. Hint: Use similar triangles to get an equation relating $$h$$ and $$r$$.) (answer)