Chapter 8

Maxima and Minima

Educators

CD

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?

Nathan W.
Numerade Educator

Problem 4

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

Hayden W.
Numerade Educator

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?

CD
Courtney D.
Numerade Educator

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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