)A square piece of cardboard is made into an open top box by cutting out squares

Question

)A square piece of cardboard is made into an open top box by cutting out squares on each corner. If the length of the sides of the cardboard is L, what should the dimensions be for the cutout so the volume is a maximum?

A lighthouse that is 1 kilometer from shore lamp rotates 8 times per minute.

What speed is the lamp%u2019s spot moving as it passes the point where it is

perpendicular to shore?

How fast is the spot moving along the shore when the spot is 1 km from

the point at which it is perpendicular?

Where is it moving the fastest and why?

4.) If you have 300 meters of fence and you desire to fence along a river (requiring no fence along it) and you have 4 fence posts (you don%u2019t have to use them all), assuming all segments of the fence have equal lengths, how should you configure the fence to get the maximum area enclosed?

5.) Suppose a Ferris wheel with radius of 12 meters is rotating at a rate of 2 rotations per minute.

How fast is a person rising when the person is 3 meters above the horizontal line running through the center (3 or 9 o%u2019clock position)?

How fast is a person rising when the person is 3 meters above the base of the Ferris wheel?

Suppose a person riding the Ferris when drops a ball while going up when they are 6 meters from the bottom of the Ferris wheel.

When will it hit the ground?

How fast will it be going when it hits the ground?

At which point should it be dropped so it is going fastest when it hits the ground?

6.) Find the following solids of revolution (rotated about the x-axis) bounded by the curves on the designated interval:

a) f (x)=sin(x)

g(x)=x

-pi<x<pi

b) f (x)=x

g(x)=sin(x*pi/ 2)

0<x<2

c) f (x)=x
g(x)=x^2-2

-4<x<4

6.) Find the following solids of revolution rotate about the y bounded by the curveson the designated interval:

a) f(x)=x^2 -x^3

x>0

x-axis

7.)A 45 degree wedge is cut from a cylinder with 1 meter radius as shown. What is its volume? Extra credit if you can solve it two ways.

8.)Find the volume remaining of a sphere of radius 5 with a hole of radius 3 drilled through its center.

Explanation / Answer

2.)

Let x = sides of square corners cut out

L-2x=sides of bottom of box

(L-2x)^2=area of bottom of box

volume of box=x(L-2x)^2

=xL^2-4Lx^2+4x^3

Let f(x)=xL^2-4Lx^2+4x^3

To find the maximum volume, you must take the first derivative of the function, set it equal to zero, then solve for x.

f'(x)=L^2-8Lx+12x^2

Now, L^2-8Lx+12x^2=0

=> L^2-2Lx-6Lx+12x^2=0

=> L(L-2x)-6(L-2x)=0

=> (L-2x)(L-6x)=0

=> L=2x, L=6x

=> x=L/2, L/6

Now, f''(x)=-8(L-3x)

At x=L/2, f''(x)=-8(L-3L/2)=+ve

At x=L/6, f''(x)=-8(L-3L/6)=-ve

=> f(x) is maximum at x=L/6

=> volume is maximum at x=L/6

3.)

Circumference that the spot travels=2*pi*(1 km)^2, 8 times per minute

1.) So speed=16pi km/min=960 km/hr

2.) Speed along the shore=2*pi*(1 km)^2*8=16pi km/min=960 km/hr

3.) It is moving with the same speed everywhere. Only if we move away from the lighthouse, its speed will SEEM to increase. (Google lighthouse paradox for more info)

4.) Let length be l and width be w. Also, the length is divided into two segments using a fence post and two other fence posts are at the corner of the rectangle (assuming no fence posts are needed along the river. if you need to change this condition contact in comment)

l+2w=300

=> 4w=300 (l=2w, since fences are of equal length)

=> w=75

So its a rectangle with width=75 ft, and length=75*2=150 ft.

5.) Suppose a Ferris wheel with radius of 12 meters is rotating at a rate of 2 rotations per minute.

circumference=2*pi*12=24pi

Now, 24pi=tangential speed*time

=> 24pi=tangential speed*1/2 (min)

=> tangential speed=48pi m/min=(4/5)pi m/sec

1.) Now the speed is based on the cos of the angle. So when the person is 3 meters above the horizontal line running through the center,

angle=arctan(3/12)=arctan(1/4)=14.04 degree

So cos(14.04 degree)=0.97

So speed=0.97*(4/5)pi=2.43 m/s

2.) When a person is 3 meters above the base of the Ferris wheel,

angle=arctan(9/12)=arctan(3/4)=36.87 degree

So cos(36.87 degree)=0.8

So speed=0.8*(4/5)pi=2.01 m/s

3.) When they are 6 meters from the bottom of the Ferris wheel, height=16-6=6 m

3.1.) So s=(1/2)gt^2

=> 6=(1/2)gt^2

=> 12/9.8=t^2

=> t=1.1 sec

3.2.) v=-gt=-9.8*1.1=10.78 m/sec

3.3.) Obviously from the highest point

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