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If c and d are inversely proportional, then c*d always stays the same. c=9 when d=8, so c*d is always 72. This means that d=72/8=9 when c=8.

Find g(5).

Hello Guest!

\(f(x) = 3x - 8=5 \\ x=\frac{13}{3}\\ f(x)=f(\frac{13}{3})=5\)

\(g(f(x)) =g(f(\frac{13}{3}))=\color{blue}g(5)= 2x^2 - 5x + 3=28\)

Frohe Weihnachten!                             Merry Christmas

There is a reason for the pattern - can you see why? Do you see anything intesting if we generalize \(x\) for not just 2, but for all integral \(x\)? Try finding a general formula. Also notice - does your 1, 0, etc. pattern have anything to do with \(x=2\)? Maybe there is, and maybe not. Try to find out. :)

This is correct. Nice job!

Perhaps, you notice in your answer that \(2115 = 46^2 - 1\)? This is no coincidence.

Instead of letting \(f(0)=0\), we let \(f(0)=1\)

And it follows that \(\sum_{i=0}^{99}f(i)=(1+1+2+3+\cdots+9)^2=46^2\)

Subtracting \(f(0)\), we have 

\(\begin{align*} 46^2 - f(0) &= 46^2 - 1 \\ &= 2116 - 1 \\ &= \boxed{2115} \end{align*}\)

Oh no, it seems like we got different answers. 

Do you mind explaining what you did?

Thanks

=^._.^=

Hi :))

I'm going to group the numbers to make them easy to calulate. 

i = 10, 20, 30, ..., 90. 1 + 2 + 3 + ... + 9

1 <= i <= 9. 1 + 2 + 3 + ... + 9 

11 <= i <= 19. 1 + 2 + 3 + ... + 9 

21 <= i <= 29. 2(1 + 2 + 3 + ... + 9)

31 <= i <= 39. 3(1 + 2 + 3 + ... + 9)

...

91 <= i <= 99. 9(1 + 2 + 3 + ... + 9)

Overall. there are 1 + 1 + 1 + 2 + 3 + ... + 9 = 2 + 10*9/2 = 47 groups of (1 + 2 + 3 + ... + 9)

47*(1 + 2 + 3 + ... + 9) = 47*45 = 2115.

I hope this helped. :))

=^._.^=

f^1(1) = (2 - 1)^2 = 1

f^2(2) = f^1(f^1(2)) = f^1(1) = 0

f^3(2) = f^1(f^2(2)) = f^1(0) = 1

f^4(2) = f^1(f^3(2)) = f^1(1) = 0

There's a pattern. :))

It seems like when f(a) = 0, f(a+1) = 1 and when f(a) = 1, f(a+1) = 0. 

So when n is odd, then the answer is 1.

If n is even, the answer is 0.

f^7(2) = 1

=^._.^=

The answer is 1+3=4

Area of equilateral triangle

 Each half of the triangle is   1/2 b * h       b = 2    h = 4 cos 60 =   4 sqrt 3/2  =    2 sqrt 3

    so each is      1/2   2    2 sqrt3 = 2 sqrt 3   

There is two of these triangles in the equilateral triangle so total area = 2 * 2 sqrt 3 = 4 sqrt 3   ( as MG 13 found)

It's asking for a ratio between the perimeter and area of an equilateral triangle with sides of 4 units. 

So, we need to calculate both perimeter and area. 

Perimeter:

4 * 3 = 12 units 

Area: 

\(a = {\sqrt{3} x^2 \over{4}}\)

\(={4\sqrt3}\)

Notice - it's asking for a ratio for Area to Perimeter. So the answer should have area in numerator, and perimeter in denominator. 

\({4\sqrt3\over12} = {\sqrt3\over3}\)

And that's your answer. 

In my solution I used the area of an equilateral equation. If you've never seen it before, research it. Google has great sources to clarify it for you. 

j = -18/5.

N = 2158.

Total number of balls = 4

Number of balls selected = 3

Then total probability = 2/4 * 1/3 = 1/6

where 2/4 represents that out of 4 balls 2 of

Same color one selected and then as the selection

is without replacement so neat balls having 

1/3 probability. of selection on 1 ball is already selected

and not kept back in bag.

1 Can of Paint to Paint 1/18 of their house.

1/18 of house paint by 1 can

1 of house Paint by 18 can

4/9 of "     "     " = 18 * 4/9 = 8 can

5/6 "         "     " = 18 * 5/6 = 15 can

They buy total (8 + 15) can of paint

But it's required 18 can to paint the

whole house.

They should return = (23 - 18) = 5 can

5 can they should return to the store.

Incorrect.... the degree will be 4 

f(x) - g(x) = -x^4 +3x^2 + 3

I got the answer its 5

I think thats wrong

Thanks for trying though i appreciate it

The degree of f(x) - g(x) is 3.

They should return 8 cans of paint to the store.

cos C = sqrt(1 - (4/7)^2) = sqrt(33)/7.

MN = 2√41

x = original money amount

p = 4m      (given)

   or   1/4 p = m

1/5 x  =  4p + 8 ( 1/4 p)

1/5 x = 6p

he has 4/5 of his money left     he spends 1/2 of this   or  4/10 x

ratio     (1/5 x) / (4/10 x) = 6p / ?

              10/20 = 6p / ?

                ? = 12        plus the original 4   = 16 pineapples

MN = sqrt ( 8²  + 10²)