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HOW DO YOU FIND THE ARC MEASURE IN A CIRCLE

I'm going to assume they gave you an arc length, and the radius of the circle.

\(\text{Let the arc length be }\ell \\ \\ \text{and let the radius be } r \\ \\ \text{then the arc measure or angle the arc subtends, }\theta \text{, is given by}\\ \\ \theta = 2 \pi \dfrac \ell r~radians = 360 \dfrac \ell r~degrees\)

This applies for

\(0 \leq \ell < r \\ \\ 0 \leq \theta < 2\pi ~radians\) 

4t-t^2-3

I'm going to assume you want to know what values of t make this zero.

\(4t-t^2-3 = 0 \\ \\ t^2 - 4t + 3 = 0 \\ \\ (t-3)(t-1)=0 \\ \\ t=3 \text{ or } t=1\) 

7 x 4 x 1 = 28

7 + 4 + 1 = 12

-9<9x+9<27

\(-9 < 9x + 9 < 27 \\ \\ -1 < x+1 < 3 \\ \\ -2 < x < 2\) 

Find all solutions of the equation in the interval [0, 2π). (Enter your answers as a comma-separated list. If there is no solution, enter NO SOLUTION.)

4 cos^3 x = 4 cos x

\(4 \cos^3(x) = 4\cos(x)\\ \\ \cos^3(x) = \cos(x)\)

\(\text{If }\cos(x) \neq 0 \\ \\ \cos^2(x)=1 \\ \\ \cos(x)=\pm 1\\ \\ x = 0, \pi\) 

\(\text{If }\cos(x)=0 \\ \\ x=\dfrac \pi 2, \dfrac {3\pi}{2}\)

so the full set of solutions is

\(x=\left \{0, \dfrac \pi 2, \pi, \dfrac {3\pi}{2}\right \}\)

How many ways are there to put 6 boxes in 3 crates if the boxes are distinguishable but the crates are not?

You can think of this as a 3 digit number in base 6.  Each digit represents how many boxes in that crate.

There are

\(6^3=216\) 

combinations, however they aren't all distinguishable.  As there are 3 crates any given distribution of the boxes into crates is indistinguishable within a group of 3! other distributions.  

For example 312 is indistinguishable from

123, 132, 213, 231, 321

So the final number is

\(\dfrac {216}{3!} = \dfrac{216}{6} = 36\)

Over a period of 120 years Bendigo produced 700 tonnes of gold.  With todat's value at around $1200 per ounce what is the value of Bendigo's gold production?

\(value = 700 ton\times ( 2,204.6lb/ton)\times (16oz/lb) \times (1200dollars/oz)=\$29,629,824,000\)

I guess the trick is to note that tonnes refers to 2000kg not lbs.

What is 2e+22's number form?

 \(2^{22} = 4194304\)

Mrs.Young's tennis court is a rectangle that is 30 feet wide by 70 feet long. What is the perimeter?

really?

\(P = 2w + 2l = 2(30)+2(70) = 60+140=200feet\) 

(5x+4) - (1-2x)

\((5x+4) - (1-2x) = (5x - (-2x)) + (4-1) = 7x+3\)