CPhill

a) a spade or club =

P(spade or club)  = P(spade) +  P(club) - P(spade and club)  =

                                   1/4        +    1/4    -       0    =

                                                 2/4  =  1/2

b) a diamond or queen

P(diamond or queen)  = P(diamond) +  P(queen) - P(diamond and queen)  =

                                          13/52       +     4/52    -            1/52   =

                                                      16/52  =   4/13 

c) a spade and then a club  =   (1/4)(13/51)  =  13/204

d) a pair of 2's  =  ( 1/13 ) (3/51) =  3/663  =  1/221

The principal, P,  must have been  

Ending Amount - Interest Earned  =

 $6044.34 -  $1344.34  =  $4700

We can find the interest rate, r, thusly :

 $6044.34 =  $4700 ( 1 + r/12)^(12 * 6)

$6044.34 =  $4700 ( 1 + r/12)^(72 )           divide both sides by $4700

6044.34 /  4700  =  ( 1 + r/12)^(72 )     take the log of both sides

log ( 6044.34 /  4700 )  = log ( 1 + r/12)^(72 )   and we can write

log ( 6044.34 /  4700 )  = 72 * log ( 1 + r/12)       divide both sides by 72

log ( 6044.34 /  4700 )  / 72  =   log ( 1 + r/12)

0.00151738  =  log ( 1 + r/12)     and we can write

10^ (0.00151738)  =  ( 1 + r/12)  subtract 1 from both sides

10^ (0.00151738) - 1  =  r/12  

0.00350001 =  r/12     multiply both sides by 12

.042  =  r  =     4.2%

h(x)  =  -343^x 

The domain is all real numbers

The range  is   ( - infinity, 0 )

As x decreases, h(x)  increases to the upper bound of ≈   0

As x increases, h (x)  decreases ......the right end behavior is towards   - infinity

Here's the graph : https://www.desmos.com/calculator/g85xk6uq8p

Mmmmm....this one looks interesting  !!!

We can let the amount after year 1 be represented as

P ( 1 + r)  = 2251.05     (1)

Where P is the orginal amount invested  and r is the interest rate

After year 2 we have

P (1 + r)^2  =  2356.85    (2)

Dividing  (2) by (1)  we have that

(1 + r)  = 2356.85 /  2251.05       subtract  1 from both sides

r  =    2356.85 /  2251.05    - 1    ≈  .047 ≈  4.7 %

We can find the original amount invested as follows

P  ( 1 + .047)  =  2251.05     divide both sides by  (1 + .047)

P  =  2251.05 / ( 1 + .047 =  $2150

We can find the time to double - the amount will be 2P - with this :

log [ (2P / P) ]  /  log ( 1 + interest rate )

log (2)  / log (1+. 047)  ≈  15.09  years

However....I'm sure  that the answer isn't 450, either....

Assuming that we can't begin with 0, there are 900 integers from 100 - 999 and half of these will be even = 450....and some of those will have repeated digits.....so.....the answer is less than 450

Here's a "brute force"  method

Let's look at all the integers that end in 0

There are  10 of these in each group of 100.....and we have 9 groups of 100 from 100-999

So....that's 10 * 9  = 90 possible integers.....but...we have to discard 10 of these (100, 200, 300. etc.) because of the repeated 0....so......that makes 80 possible numbers ending in 0

Now.....let's look at all the numbers ending in 2

Again there are 10 of these in each group of 100.....so, again, that's 90 possible numbers

But.....from these we must subtract all the numbers that have 2 as a middle digit and 2 as an ending digit.....and there are 9 of these   (122, 222, 322, 422, etc )...so that gives us 81 possible numbers....

Also.....we have to subtract the numbers that start with 2 and have 2 as an ending digit....there are 9 of these, as well  (202, 212, 232, 242, etc. ).....(note that we have already counted '222")....so.....the possible numbers ending in 2 with no repeated digits  = 90 - 9 - 9  =  72

And the other numbers ending in 4,6 or 8  will similarly occur 72 tmes each

So.....the total combinations ending in an even number with no repeated digits  = 80 +  4 * 72  =

80 + 288  =   368

As Melody always says, "That's what I think!!! "

Sorry about that....I mis-read the question.....

There is no real solution to this

Squaring both sides, we have that

t + 4  =  4

Subtract 4 from both sides

t  = 0

But.......subbing this back into the original equation means that

√] 0 + 4]  =  -2

√4  =  -2

2  = -2

Which is false

If the ships' paths are perpendicular to each other, the angle between them can't  be 47°

16  =  .40 * .50 * N          simplify

16  =  .20 * N                 divide both sides by  .20

80 = N =  the  number

We have 10 ways to choose the first number, 9 ways to choose the second and 8 ways to choose the third

So

10 * 9 * 8   =    720  possible combinations