hectictar

I think there are 10 different combinations that will satisfy Bill's order requirements.

Let's say the different kinds of donuts are  A, B, C,  and  D.

We know that Bill is always going to order A, B, C,  and  D, so that just leaves  2  unused orders ( 6 - 4 = 2 ).

To finish his orders, Bill can choose one of the following possibilities:

AA

AB

AC

AD

BB

BC

BD

CC

CD

DD

There are 10 different ways he can finish his orders, so that means there are 10 different combinations that will satisfy Bill's order requirements.

There are 2 values of  x  within  0° ≤ x ≤ 360°  that satisfy  sin x  =  -0.73

They are:

x  ≈  226.886°

x  ≈  313.114°

The tip of the hour hand travels the circumference of its circle once every 12 hours.

24 hours / 12 hours  =  2     So the hour hand travels its circumference 2 times per day.

The tip of the minute hand travels the circumference of its circle once every 1 hour.

24 hours / 1 hour  =  24     So the minute hand travels its circumference 24 times per day.

circumference of hour hand circle  =  2π( 5 cm )  =  10π cm

circumference of minute hand circle  =  2π( 10 cm )  =  20π cm

distance traveled by tip of hour hand  =  2 * 10π cm  =  20π cm

distance traveled by tip of minute hand  =  24 * 20π cm  =  480π cm

20π cm + 480π cm   =   500π cm   =   5π  meters   ≈   15.708 meters

990° / 180°  =  5.5

So there are only  5  values of  x  within  0° ≤ x < 990°  that satisfy  sin x  =  -0.31

They are:

x  ≈  198.059°

x  ≈  341.941°

x  ≈  558.059°

x  ≈  701.941°

x  ≈  918.059°

\(\sum\limits_{i=3}^5i^{(i+2)}\ =\ 3^{(3+2)}+4^{(4+2)}+5^{(5+2)}\\~\\ \phantom{\sum\limits_{i=3}^5i^{(i+2)}}\ =\ 3^5+4^6+5^7\\~\\ \phantom{\sum\limits_{i=3}^5i^{(i+2)}}\ =\ 243+4096+78125\\~\\ \phantom{\sum\limits_{i=3}^5i^{(i+2)}}\ =\ 82464\)_

1234  +  4321   =   5555

2345  +  5432   =   7777

3456  +  6543   =   9999

4567  +  7654   =   12221

5678  +  8765   =   14443

6789  +  9876   =   16665

And....

gcd(5555, 7777, 9999, 12221, 14443, 16665)  =  101 * 11  =  1111

Here is another diagram:

The person is sitting at point  B .

height of person  =  CD + DE

                                                And  DE = 1  because the bottom of the wheel is 1 m off the ground.

height of person  =  CD + 1

Now we just have to find the length of  CD .

CD  =  AD - AC

                               And we know  AD = 8  because the radius of the Ferris wheel is  8 m.

CD  =  8 - AC

And we can find the length of AC by looking at △ABC.

cos θ  =  adjacent / hypotenuse

cos θ  =  AC / AB

                               And we know  AB  =  8  beause the radius of the Ferris wheel is  8 m.

cos θ  =  AC / 8

                               Multiply both sides of the equation by  8 .

8 cos θ  =  AC

So........

height of person   =   CD + DE   =   (AD - AC) + DE   =   ( 8 - 8 cos θ ) + 1  =  -8 cos θ + 9

And....if you don't understand how to relate  θ  and  t .....this might help: try to imagine what would that angle be after 10 seconds? (10 seconds is half of 20 seconds!) What would that angle be after 5 seconds? (5 seconds is a fourth of 20 seconds.)

Imagine you are a cashier and need to give a customer back some change, and you want to return the fewest number of coins to the customer as possible. For example, if the change due was 50 cents, you would never return ten 5 cent coins when you could just return one 50 cent coin.

To ensure that you return the fewest coins possible, you might think through these steps in order:

1. Return the maximum number of 50 cent coins that you possibly can.

2. Return the maximum number of 20 cent coins that you possibly can.

3. Return the maximum number of 10 cent coins that you possibly can.

4. Return the maximum number of  5 cent coins that you possibly can.

nth term  =  3 * 5^{(n - 1)}

Also this might help to visualize a way that we could accomplish this:

So the total number of connections  =  6 * 5  =  30