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Chapter 7: Q. 100 (page 497)

A rectangle is inscribed in a semicircle of radius 1. See the illustration.

Part (a): Express the area A of the rectangle as a function of the angle 胃shown in the illustration.

Part (b): Show that A胃=sin2胃.

Part (c): Find the angle 胃 that results in the largest area A.

Part (d): Find the dimensions of this largest rectangle.

Short Answer

Expert verified

Part (a): Area A as a function is given by A胃=2cos胃脳sin胃.

Part (b): To prove A胃=sin2胃use the double angle formula and solve the equations.

Part (c): The area Ais largest at 胃=45掳.

Part (d): The dimension of largest rectangle is1unit脳1unit.

Step by step solution

01

Part (a) Step 1. Write function of area.

Consider the given question,

Radius of circle is 1unit.

x is the half width and y is the length of the rectangle.

localid="1646410018881" length=2脳1cos胃breadth=1sin胃

We know area of rectangle islocalid="1646410062384" A=length脳breadth.

Substituting the values in the formula,

localid="1646410078022" A胃=2cos胃脳sin胃

02

Part (b) Step 1. To prove Aθ=sin2θ.

Consider the previous part,

A胃=2cos胃脳sin胃......(i)

From the double angle formula,

sin2胃=2cos胃脳sin胃......鈥�(ii)

From equations (i) and (ii),

A胃=sin2胃

Hence, proved.

03

Part (c) Step 1. When the angle results in largest area.

Consider the previous part,

A胃=sin2胃

For the area to be the largest, maximum value of sin2胃. Then,

sin2胃=12胃=90掳胃=45掳

04

Part (d) Step 1. Find the dimensions of this largest rectangle.

Consider the previous part,

A胃=sin2胃

Then,

localid="1646411213709" A=sin90掳=1

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Most popular questions from this chapter

Projectile Motion A golfer hits a golf ball with an initial velocity of 100 miles per hour. The range R of the ball as a function of the angle 胃to the horizontal is given by R(胃)=672sin2胃, where R is measured in feet.

(a) At what angle 胃 should the ball be hit if the golfer wants the ball to travel 450 feet (150 yards)?

(b) At what angle 胃 should the ball be hit if the golfer wants the ball to travel 540 feet (180 yards)?

(c) At what angle 胃 should the ball be hit if the golfer wants the ball to travel at least 480 feet (160 yards)?

(d) Can the golfer hit the ball 720 feet (240 yards)?

Rewrite in terms of sine and cosine functions: cot(胃)sec(胃)

Projectile Motion The range R of a projectile propelled downward from the top of an inclined plane at an angle 胃to the inclined plane is given by

R(胃)=2v02sin鈦�胃cos鈦�(胃鈭�蠒)gcos2鈦�蠒

where vois the initial velocity of the projectile, 鈭�is the angle the plane makes with respect to the horizontal, and g is the acceleration due to gravity.

(a) Show that for fixed voand 鈭�the maximum range down the incline is given by

Rmax=v02g(1鈭�sin鈦�蠒)

(b) Determine the maximum range if the projectile has an initial velocity of 50meters/second, the angle of the plane is 鈭�=35o, and g=9.8 meters/second2.

Projectile Motion The horizontal distance that a projectile will

travel in the air (ignoring air resistance) is given by the equation

R(胃)=v02sin2胃g

where v0is the initial velocity of the projectile, 胃is the angle

of elevation, and g is acceleration due to gravity (9.8 meters

per second squared).

(a) If you can throw a baseball with an initial speed of 34.8 meters per second, at what angle of elevation 胃should you direct the throw so that the ball travels a distance of 107 meters before striking the ground?

(b) Determine the maximum distance that you can throw the ball.

(c) Graph R=R(胃),with v0=34.8meters per second.

(d) Verify the results obtained in parts (a) and (b) using a graphing utility.

Movie Theater Screens Suppose that a movie theater has a screen that is 28feet tall. When you sit down, the bottom of the screen is 6feet above your eye level. The angle formed by drawing a line from your eye to the bottom of the screen and your eye and the top of the screen is called the viewing angle. In the figure, 胃is the viewing angle. Suppose that you sit xfeet from the screen. The viewing angle 胃is given by the function:

胃x=tan-134x-tan-16x

(a) What is your viewing angle if you sit 10feet from the screen? 15feet? 20feet?

(b) If there is 5feet between the screen and the first row of seats and there is 3feet between each row, which row results in the largest viewing angle?

(c) Using a graphing utility, graph

胃x=tan-134x-tan-16x.
See all solutions

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