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Chapter 8

Maxima and Minima


Problem 1

A rectangle has its base on the $x$ -axis and its two upper corners on the parabola $y=12-x^{2} .$ What is the largest possible area of the rectangle?

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Problem 2

An open rectangular box is to be made from a $9 \times 12$ inch piece of tin by cutting squares of side $x$ inches from the corners and folding up the sides. What should $x$ be to maximize the volume of the box?

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Problem 3

A 384-square-meter plot of land is to be enclosed by a fence and divided into two equal parts by another fence parallel to one pair of sides. What dimensions of the outer rectangle will minimize the amount of fence used?

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Problem 4

What is the radius of a cylindrical soda can with volume of 512 cubic inches that will use the minimum material?

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Problem 5

A swimmer is at a point 500 m from the closest point on a straight shoreline. She needs to reach a cottage located 1,800 m down shore from the closest point. If she swims at 4 m/s and she walks at 6 m/s, how far from the cottage should she come ashore so as to arrive at the cottage in the shortest time?

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Problem 6

Find the closest point on the curve $x^{2}+y^{2}=1$ to the point $(2,1)$

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Problem 7

A window consists of an open rectangle topped by a semicircle and is to have a perimeter of 288 inches. Find the radius of the semicircle that will maximize the area of the window.

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Problem 8

The range of a projectile is $R=\frac{v_{0}^{2} \sin 2 \theta}{g},$ where $v_{0}$ is its initial velocity, $g$ is the acceleration due to gravity and is a constant, and $\theta$ is its firing angle. Find the angle that maximizes the projectile's range.

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Problem 9

A container with a square base, vertical sides, and an open top is to be made from 1000 $\mathrm{ft}^{3}$ of material. Find the dimensions of the container with the greatest volume.

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Problem 10

Where on the curve $y=\frac{1}{1+x^{2}}$ does the tangent line have the greatest slope?

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