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These are the squares of ALL prime numbers below 100:

(4, 9, 25, 49, 121, 169, 289, 361, 529, 841, 961, 1369, 1681, 1849, 2209, 2809, 3481, 3721, 4489, 5041, 5329, 6241, 6889, 7921, 9409)

As you can see, there isn't a single one with its units digit = 7. Therefore, the probability is zero!.

If the link doesn't work for you then let me know.

Your part a is correct.

I made this pic to help you understand what you are finding.

You are finding gradients.

A gradient = rise on run = difference in y values over difference in x values.

OMG you're back!!! :)

3  12  21  30  102  111  120  201  210  300  1002  1011  1020  1101  1110  1200  2001  2010  2100  3000  Total =  20 such numbers.

Say the side length of hexagon = 1, so perimeter = 6. The circumference of circle must be 6, so the radius is 2 * pi * r = 6 and r = 3/pi.

The area of the hexagon is 3sqrt(3)/2 by the formula (or draw 6 equliateral triangles). The areae of circle is 9/pi.

The ratio of hexagon to circle is (3sqrt(3)/2)/(9/pi) = (3sqrt(3) * pi) / (18) = (sqrt(3) * pi) / 6.

You are very welcome!

:P

The arc is 120/360 = 1/3 around the circle. So the chord makes a triangle with 30-30-120 angles and sides radius, radius, 8 in.

Split this in the middle of the 120 angle to form two 30-60-90 triangles. The side lengths for the triangle are 4/sqrt(3), 4. The area is 8/sqrt(3), and since there are two the area is 16/sqrt(3) = 16sqrt(3)/3.

You are very welcome!

:P

Teh largest 3-digit base 10 number is 999. You take the biggest number and put it in all values.

The largest 3-digit base 14 integer is DDD. This in base 10 is 14^2 * 13 + 14 * 13 + 13 = 2743.

You are very welcome!

:P

45 > 12t + 3t - 24 - 6

45 > 15t - 30

75 > 15t

5 > t

You are very welcome!

:P

Hi thank you for the reply! 

f(x)=2x^3 - x^2 - 6x+12

a.f(12)=2(12)^3 - (12)^2 - 6(12)+12 = 3252

f(0)=2(0)^3 - (0)^2 - 6(0)+12 = 12

3252-12 = 3240

3240/12 = 270

So the average change is just 270.0?

c. f(3)=2(3)^3 - (3)^2 - 6(3)+12 = 39 

f(3.001)=2(3.001)^3 - (3.001)^2 - 6(3.001)+12 = 39.042017002

39 - 39.042017002 = -0.042017002

-0.042017002/-0.042017002 = 1

I'm guessing I'm doing something wrong here and misunderstanding your directions at a. ? 

As follows:

Let f(g(x)) = 3x + 3 and f(x) = x + 6. If g(x) = ax + b, compute a + b.

start with this

f(x)      = x + 6.           and    g(x) = ax + b

f(g(x)) = g(x)+6

so

f(g(x)) = ax+b+6

which means

3x+3=ax+(b+6)

now just equate coefficients.     :)

I don't really know how to work with intervals so:

Given is the function f(x)=2x^3 - x^2 - 6x+12

a. Calculate to one decimal place the average change of the function over the interval [0, 12]

You are being asked to find    f(12)  - f(0)

And then to divide the answer by 12-0

b. Given is that the differential quotient of f on the interval [0, a] equals 0. How do I calculate a?

Same as above only this time the answer is 0

c. Approximate the slope of the graph of f at the point with x-coordinate 3. Let Δx be 0.001    

Same as (a)  only using   the interval [3, 3.001]

d. On the interval [(-1/2) (1/2] there is a point P on the graph of f where the slope of the tangent line is the smallest. Determinte the coordinates of P to one decimal place and please explain what you did.

You can think about this one after you do the others.

I suggest you graph your function so that you can understand what you have done.

If you do some yourself, and display it, then you can ask more guidance if you want to.

Obviously the sides have to add to 20 and they have to be integers.

the only other rule is that the 2 little sides have to add to be more than the length of the 3rd side

The longest one cannot be 19 the two little ones cannot add to 19

Can the longest one be 18?

work your way down and work out what triples can work.

Give it a go yourself  :))

Thanks Alan   

The ¬ symbol is the logical negation symbol Melody. It means NOT(whatever follows). 

Okay...

*First I would like to say that I JUST REALIZED THAT THE PLACE WAS AOPS-LANDIA. :)*

1. The cyclist has traveled on the hilly road to Beast Island from Aopslandia and then back to Aopslandia, which means that the cyclist went uphill once at a rate of 20km per hour and went downhill once at a rate of 30km per hour.

2. Let the distance between Aopslandia and Beast Island be D (kilometers), and let the time needed to travel from Aopslandia to Beast Island be T1 (hours). Let the time needed to travel back from Beast Island to Aopslandia be T2 (hours).

3. The formula for average speed is Total Distance divided by Total Time. In this case, the total distance is D+D=2D, and the total time is T1+T2 hours, so the average speed will be (2D)/(T1+T2). Time equals distance over speed, so we can rewrite the expression into (2D)/(D/20)+(D/30). The least common multiple of 20 and 30 is 60, so the expression will turn into (2D)/(3D/60+2D/60)=(2D)/(5D)/60. Let's apply the fraction rule a/b/c=(a*c)/b turning the expression into (2D*60)/(5D)=(120D)/(5D). 120/5=24, so the cyclist's average speed during the entire round trip is 24km per hour.

Answer: 24km per hour 

I meant "A cyclist travels at 20 kilometers per hour when cycling uphill, 24 kilometers per hour when cycling on flat ground, and 30 kilometers per hour when cycling downhill. On a sunny day, they cycle the hilly road from Aopslandia to Beast Island before turning around and cycling back to Aopslandia. What was their average speed during the entire round trip?"

Your diagram is not drawn correctly!!!

CE = ED' = 3

Find angle CED'

Use the law of cosines to find the length of CD'

Warning: if you do not even attempt to solve this, I'll delete my post. 

I know the answer, but I won't tell you.

What is the little symbol at the beginning?

¬(x>-2) = ?

Thannk you 

That is a really neat proof!   Thanks guest.

ok, thanks   But I would prefer to describe it as

11! /(5! * 3! * 3!)   divide by 2 because the 3! and 3! are interchangeable. multiply by 2 because there are 2 ways to add in Henry.

=  9,240 ways

I am still having some trouble thinking about it this way though. 

This would normally be the answer to a question like this.

How many ways can the 11 letters  aaabbbcccc be arranged in a line.   11!/(3!3!4!)  is the answer.    [I am not going to concern myself with Henry]

For our question it would be how many ways can 11 objects be put in a line if the order of the first 5 does not matte, the order of the next 3 does not matter and the order ot the last 3 does not matter AND the second and third groups of 3 can be swapped around. 

NOW it does make sense to me.    11!/(3!3!4!) divided by 2

Thanks guest for making me think about that.

For all the letters in the 4 letter word you can choose from 7 different letters. If we have 4 slots and 7 choices then we get \(4\cdot 7 = \boxed{28}\)