Yeah...even if Alan and I disagree about how long it takes for 62 apples to fall off a tree.....I was assuming that the first apple was "on the fence" about the whole thing and it took him/her almost 3 hrs. to decide to take the "plunge"  ...........either way, it's either 7.2 days or 7.3 days.......depending upon your local time zone and prevailing gravitational conditions, of course.......!!!

Yes you have your clever moments very often.   

Well....I DO have my "moments".......(every now and then) (LOL!!!)

It was cluey for you to notice that Chris.     

Complementary angles are angles that add up to 90.

Complementary trig ratios are ones where the values are equal for complementary angles.

I am probably not wording that very well.

Cosine stands for complementary ratio of sine   (at least that is how I think of it)

     Cosine(theta)=sine(90-theta)

Cotangent stands for complementary ratio of tan

     Cot(theta)=tan(90-theta)

Cosec stands for complementary ratio of sec

     cosec(theta)=sec(90-theta)

If you remember this it can help a lot.    

We don't really need to use any calculators here.......

Notice that cosA = sin(90 - A)

So

cos 12  = sin(90 - 12)

cos12  =  sin 78

And, if like things are subtracted, the answer  = 0 !!!!

y=cx

substituting you get

$$\\11=c\times 18\\\\
c=\frac{11}{18}\\\\
y=\frac{11}{18}\times x$$

Now just substitute in different values of x to get the corresponding values of y  

x - 3(x +5) = -(3x + 20)      let's simplify this first

x - 3x - 15  = -3x - 20         combine like terms

-2x - 15  = -3x  - 20            add 3x to both sides  and add 15 to both sides

x = -5

Feel free to use the calculator on the home page. 

x-3(x+5)=-(3x+20)

$$\\-3(x+5)=-3x+-15=-3x-15\\\\
-(3x+20)=-1(3x+20)=-3x+-20=-3x-20\\\\
so\\
x-3(x+5)=-(3x+20)\\
x-3x-15=-3x-20\\
-2x-15=-3x-20$$

And you can take it from there.

Notice that if you multiply a bracket by a negative number, everything inside the bracket changes sign. 

Melody is correct, rosala......"visualizing" graphs often takes some experience....this one isn't too bad, however.....

When we have something that has the form  y = mx ± b, this is always a graph of a line ...

We don't even need to make a "table" in this case.....just pick two values for "x" and find their corresponding "y's".......this will give us two points on the graph....then, we only need to "connect the dots"

For instance, if we let x = 0 in the above equation y then  = 1. And if we let x = 1, then y = 3. So we have two points, (0, 1) and (1, 3). All we need to do is to plot these on the graph and draw a line between them......voila !!!.....there's our graph  !!!  Check out the graph I generated in Desmos in the web link...you'll see that those two points are on the graph !!!

BTW ......you might want to give Desmos a try......it's easy to use, and you can get some practice in recognizing some different graphs just playing around with it......!!! Also, some advanced courses like calculus enable you to visualize some pretty complicated stuff by using a technique known as "curve sketching" .....(you'll get to that later, of course !!!)

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Use mod(x,y) or x mod y (It can be entered in either format).  For example:

$${\mathtt{13}} \,{mod}\, {\mathtt{4}} = {\mathtt{1}}$$

Looks like f(x) = -1- x²

               g(x) = sin(pi*x/2)

It depends on what information you have.  If you have two (x, y) points then it is the difference between the y values divided by the difference between the corresponding x values.

Set W(t) = 10³ and solve for t.

10³ = 6*10¹⁰*e^-0.000122t  You didn't have an "e" in your expression, but I suspect it should be there, so I've included it.

Divide both sides by 6*10¹⁰

10^-7/6 = e^-0.000122t  

Take logs (natural log) of both sides:

ln(10^-7/6) = -0.000122t

Divide both sides by -0.000122

t = -ln(10^-7/6)/0.000122 = (7*ln(10)+ln(6))/0.000122

$${\mathtt{t}} = {\frac{\left({\mathtt{7}}{\mathtt{\,\times\,}}{ln}{\left({\mathtt{10}}\right)}{\mathtt{\,\small\textbf+\,}}{ln}{\left({\mathtt{6}}\right)}\right)}{{\mathtt{0.000\: \!122}}}} \Rightarrow {\mathtt{t}} = {\mathtt{146\,802.091\: \!149\: \!068\: \!645\: \!811}}$$

t ≈ 146802 years

If you start timing from the instant the first apple falls off then the time is 170*61mins because there are only another 61 left to fall!

$${\mathtt{t}} = {\mathtt{170}}{\mathtt{\,\times\,}}{\mathtt{61}} \Rightarrow {\mathtt{t}} = {\mathtt{10\,370}}$$ minutes

t ≈ 7.2 days

Yes Rosala, 

there is a jump in understanding between working out equations and seeing how they relate to graphs.

Just fill out  table of values like this one and plot the points, 

 That is a good place to start.

Do you understand what I mean?

$$\begin{tabular}{|c|c|c|c|c|}\hline
x&-1&0&1&2 \\\hline
y&&&&& \hline


\end{tabular}$$

When you joint them up they should make a line because equations that look like this always plot to lines.

You will slowly learn to recognise what a lot of different graphs of equations will look like.  It is really quite cool when you get the hang of it.

Cphill where did u find that!

i cant understand graphs at all!they seem of no use to me!and even confusing aswell!

Melody why cant i understand your words!is this hello really!

anyways just kidding!

hello Melody!

Hello Rosala

that is becoz .......................melody and i are talking in our secret language!

6/3=2

So the most simple exact answer is just

$${\mathtt{2}}{\mathtt{\,-\,}}{\sqrt{{\mathtt{13}}}}$$

(3√X multipled by 3√X^4)/ X^ -1/3

It is not clear what is rasised to the power of 4   or if the x at the end is raised to -1 or to -1/3

YOU NEED TO USE MORE BRACKETS.

$$(3\sqrt X \times 3\sqrt{ X^4})/ X^ {-1/3}$$

$$\\(3X^{1/2} \times 3X^2)/ X^ {-1/3}\\\\
=(9X^{1/2} \times X^2)/ X^ {-1/3}\\\\
=9X^{1/2+2-(-1/3)} \\\\
=9X^{(5/2)+(1/3)} \\\\
=9X^{(15/6)+(2/6)} \\\\
=9X^{(17/6)} \\\\$$