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Chapter 3: Q. 28 (page 288)

Use optimization techniques to answer the questions in Exercises 25鈥�30.
Find the area of the largest rectangle that fits inside a circle of radius 4.

Short Answer

Expert verified

The area of the rectangle is 32.

Step by step solution

01

Step 1. Given Information.

Radius of circle is 4.

02

Step 2. Form an equation.

Let x be the length of the given rectangle.

From the given information,

Diameter,=4脳2

=8

03

Step 3. Use Pythagoras theorem to find x.

Using Pythagoras theorem,

x2+x2=822x2=64x2=32

04

Step 4. Find the area of rectangle.

The area of the rectangle which fits inside the area of the circle is the area of the square.

x2=32

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Most popular questions from this chapter

Prove that the lateral surface area of a right circular cone

is equal to 蟺rl, where r is the radius of the cone and

l is the length of the diagonal of the cone, that is, the

distance from the vertex of the cone to a point on its

circumference.

Use Rolle鈥檚 Theorem to prove that if fis continuous and differentiable everywhere and has three roots, then its derivative f has at least two roots.

Q. True/False: Determine whether each of the statements that follow is true or false. If a statement is true, explain why. If a statement is false, provide a counterexample.

(a) True or False: Every local maximum is a global maximum.

(b) True or False: Every global minimum is a local minimum.

(c) True or False: If f has a global maximum at x = 2 on the interval (-鈭�,鈭�), then the global maximum of fon the interval [0, 4] must also be at x = 2.

(d) True or False: Iff has a global maximum at x = 2 on the interval [0, 4], then the global maximum of f on the interval (-鈭�,鈭�) must also be at x = 2.

(e) True or False: If f is continuous on an intervalI, then f has both a global maximum and a global minimum on I.

(f) True or False: Suppose f has two local minima on the interval [0, 10], one at x = 2 with a value of 4 and one at x = 7 with a value of 1. Then the global minimum of fon [0, 10] must be at x = 7.

(g) True or False: If f has no local maxima on (-鈭�,鈭�), then it will have no global maximum on the interval [0, 5].

(h) True or False: Iff'(3) =0, then f has either a local minimum or a local maximum at x = 3.

Determine whether or not each function f in Exercises 41鈥�48 satisfies the hypotheses of Rolle鈥檚 Theorem on the given interval [a, b]. For those that do, use derivatives and algebra to find the exact values of all c 鈭� (a, b) that satisfy the conclusion of Rolle鈥檚 Theorem.

fx=lnx2-1,a,b=-2,2

Find the possibility graph of its derivative f'.
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