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Hey there!

You can find a circles circumference by multiplying its diameter (or 2*radius) by pi!!

You got this! Hope this helped :)

There are a couple of similar right triangles: APC and BQC

AB = 5          AP = 1          AC = x             BQ = 3             BC = 5 - x

AP / AC = BQ / BC

1 / x = 3 / (5 - x)

4x = 5 

x = 1.25

AC = 1.25            BC = 3.75 

PC = sqrt(AC² - AP²) = 0.75

QC = (PC * BQ) / AP         QC = 2.25

PQ = PC + QC    

PQ = 3 cm

This is not hard.  It is combinatory maths.  Use C on your calculator.

How many ways can you choose 4 from 5?

How many ways can you choose  8 from 10?

Multiply the answers.

Others: Please let guest answer for his/her self.

If \(f(x) = y\), for 2 numbers x and y, then \(f^{-1}(y)=x\).

Now that we know that, we can find the value of \(f^{-1}(4)\), which is just 3.

Also, \(f(5) =1\) (pretty straightforward).

Can you take it from there?

By #2 Jen ate less than 4, and by #5 Jen ate more than 2, therefore Jen ate 3 slices.  

By #3 John ate less than 3 so he ate either 0, 1, or 2. 

Kev ate > 1 so a minimum of 2 but there's no upper limit.

Em ate > 3 so a minimum of 4 but there's no upper limit. 

There's no way to determine the size of the pizza.    

_.

(a) If 12, x, 24 is a Shonk sequence, what is the value of x?      

I'll take this one cuz it's the easiest.  We see that 12 • 24 is 288, which can be written as 2 • 144.

The 144 is a square so we need to find a number that when multiplied by the 2 becomes a square. 

Note that 36 is a square and half of it lies between 12 and 24, so let's use 18. 

Check:  12 • 18 • 24 = 5184   ....   and 5184 is a square, namely 72².  

_.

$1865593=0.39(x)$

$x=\frac{1865593}{0.39}$

$\boxed{x \approx 4783572}$

There were 14 total volunteers, with an average of $\frac{1306}{5*14}=\frac{1306}{70} \approx \boxed{19}$ per volunteer per day.

could you explain?

What's that?

M = 1.5 A      This is given....re arrange

M/1.5 = A

7 A + 2 M = 25       this  si given

7 ( M/1.5) + 2 M = 25

6 . 66667 M = 25

M = 3.75         

She has 75 % of her money left     80 % of this is   . 8 * 75 = 60

60 / 3.75 per mango = 16 mangoes

If he needs 1/3 of a banana for each smoothie, he needs a total of 1/3*30 bananas. Because he has 12, 12>10, so he has enough.

Can you show your work?

Sixteen more mangoes.

You didn't stae that the sides were integer lengths....

   we could build a rectangle with sided   .009   and 11.991     ( make one side as small as possible....approaching zero)

      as the small side approaches zero , the diagonal approaches a limit  of 12 units

perimeter = 2 x length + 2 x width   or  2(length + width)

and since the perimeter is 24 inches:
24 = 2 x length + 2 x width

12 = length + width

diagonal length = \(\sqrt{l^2 + w^2}\), when l = length and w = width, according to pythagoras' theorem.

since they are both squared, we want a big number (example: 2² + 3² < 1² + 4²)

this means to maximize the diagonal, the length is 11 and width is 1. (11 + 1 = 12)

\(\sqrt{11^2 + 1^2} = \sqrt{122} ≈ 11.06\)

hope this helps; tell me if I did anything wrong :)

 (a + (2n + 1)d)^2 - (a + (2n)d)^2 + ... + (a + d)^2 - a^2 = (a + (2n + 1) d + (a + 2nd + ... + a + d) + a = n(3a + (n + 2)d).

A 1 x 14 rectangle would yield a diagonal of  sqrt 197

$P(1,2,3)=\frac{1}{2+3}=\frac{1}{5}$

$P(2,3,1)=\frac{2}{3+1}=\frac{2}{4}=\frac{1}{2}$

$P(3,1,2)=\frac{3}{1+2}=\frac{3}{3}=1$

$P(\frac{1}{5},\frac{1}{2},1)=\frac{\frac{1}{5}}{\frac{1}{2}+1}=\frac{1}{\frac{15}{2}}=\boxed{\frac{2}{15}}$

By Hamilton's Theorem, the solutions are z = 4^{1/4}*e^(pi*i/4), 4^{1/4}*e^(pi*i/4 + pi/4), 4^{1/4}*e^(pi*i/4 + 2*pi/4), and 4^{1/4}*e^(pi*i/4 + 3*pi/4).  Since 4^{1/4} = sqrt(2) and e^(pi*i/4) = (1 + i)/sqrt(2), the first solution is 1 + i.  Then the other roots work out as

4^{1/4}*e^(pi*i/4 + pi/4) = 1 - i,

4^{1/4}*e^(pi*i/4 + 2*pi/4) = -1 - i, and

4^{1/4}*e^(pi*i/4 + 3*pi/4) = -1 + i.

By Hamilton's Theorem, the solutions are z = 4^{1/4}*e^(pi*i/4), 4^{1/4}*e^(pi*i/4 + pi/4), 4^{1/4}*e^(pi*i/4 + 2*pi/4), and 4^{1/4}*e^(pi*i/4 + 3*pi/4).  Since 4^{1/4} = sqrt(2) and e^(pi*i/4) = (1 + i)/sqrt(2), the first solution is 1 + i.  Then the other roots work out as

4^{1/4}*e^(pi*i/4 + pi/4) = 1 - i,

4^{1/4}*e^(pi*i/4 + 2*pi/4) = -1 - i, and

4^{1/4}*e^(pi*i/4 + 3*pi/4) = -1 + i.

$2*4=2^4+4^2=16+16=\boxed{32}$

Plugging in, you get $\frac{2}{12}+\frac{12}{3}+\frac{3}{2}=\frac{1}{6}+4+\frac{3}{2}=4+\frac{1}{6}+\frac{9}{6}=4+\frac{10}{6}=4+\frac{5}{3}=\boxed{\frac{17}{3}}$

c(2,12,3) = \(\frac{2}{12} + \frac{12}{3} + \frac{3}{2}\)

               = \(\frac{1}{6} + \frac{24}{6} + \frac{9}{6}\)

               = \(\frac{34}{6}\)

               = \(\frac{17}{3} \) or \(5\frac{2}{3}\)

Thanks! Its correct!! :D