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We want to solve this

270  =  85+​(25+7​)w     subtract  85  from both sides

185 =  32w         divide both sides by 32

w ≈ 6 weeks

Will do the scale first:

18/3 x 4 = 24 feet - diameter of the base of the fountain.

24 /2 = 12 feet - the radius of the base of the fountain.

Area of circle =r^2 x Pi, where r =radius

Area =12^2 x 3.141592

Area =452.4 Square feet - the actual area of the base of the fountain.

2x + 3y   = 11    →  3y  = 11 - 2x        (1)

3x + 3y   = 18      (2)

Put (1) into (2)    and we have

3x  + 11 - 2x  =  18        subtract  11 from both sides and simplify

x =  7          and using  (2)  to find y, we have that

3 (7)  + 3y  = 18

21  + 3y = 18     subtract 21 from both sides

3y   =  - 3       divide both sides by 3

y  =  -1 

please I really need help

Wow, thanks for the extra effort, doing it on paper like that!!

Really appreciate it!

-2/5x-9 < 9/10    add 9 to both sides

-2/5x < 9+9/10

-2/5x < 99/10     Multiply both sides by - 5/2 and reverse the inequality sign:

x > 99/10 x - 5/2

x > - 495 / 20     Reduce the RHS by dividing by 5

x > - 99 /4

(-3x + 1) / (x - 2) = -3    Cross multiply

(-3x +1) = -3(x - 2)

-3x + 1 =-3x + 6

-3x + 3x =6 - 1

0 = 5  ???!!!   NO SOLUTION !!!!

They are called "Triangular Numbers". Go online to this page and see how and why they are called that:  https://www.mathsisfun.com/algebra/triangular-numbers.html

$-\frac{2}{5}x-9<\frac{9}{10}$

First, add 9 to both sides.

$-\frac{2}{5}x<\frac{9}{10}+9$

Simplify

$-\frac{2}{5}x<\frac{99}{10}$

Divide both sides by -2/5

$x<\frac{\frac{99}{10}}{-\frac{2}{5}}$

Anything divided by a fraction is the same as that number multiplied by the fraction's reciprocal.

$x<\frac{99}{10}*-\frac{5}{2}$

99*-5=-495

10*2=20

$x<\frac{-495}{20}$

Reduce.

$\frac{-495}{20}=\frac{-99}{4}$

That is the furthest it can simplify, so x>-99/4. If you want that in decimal form, here it is.

$\frac{-99}{4}=-24.75$

Thanks Tiggsy , will do  :)

Hi Melody.

You need to read up on De Moivre's theorem.

The basic theorem says that one value of $\displaystyle (\cos \theta+i\sin\theta)^{n} \text{ is }\cos n\theta+i\sin n\theta.$

That's easy to prove when n is an integer, just use induction.

The  'one value of ', (because the angle is multivalued), becomes important when n is a fraction.

The basic result above could be expressed as

$\displaystyle \{\cos(\theta+2k\pi)+i\sin(\theta + 2k\pi)\}^{n}=\cos(n\theta+2kn\pi)+i\sin(n\theta+2kn\pi)$.

k is any integer, and if n is also an integer, the 2kpi bit isn't needed, but if n is a fraction, (the theorem still holds if this the case), you now have fractional parts of 2pi, and therefore multiple values for the roots.

For square roots the numbers are 180 degees apart so there are just two possibles, cube roots are 120 degrees apart so there are three possibles, and so on.

Tiggsy

The number of fish in a river is expected do decrease 5% every year for the next 4 years. There is currently 200 fish. How many fish will be in the river after 4 years?

200*0.95^4 = 162.90125

about 163 fish after 4 years.

Thanks Tiggsy

You lost me when you halved everything. I mean when you found the square root by halving 270 etc.

Why did you do that? Why does it work? 

Whenever you have to deal with roots or big powers of complex numbers, it's usually best, and is more often than not essential,  to put the number into polar form and to make use of de'Moivre's theorem.

$\displaystyle -i \equiv \cos(270+k.360)+i\sin(270+k.360)$, k an integer, (or zero), angles in degrees,

so

$\displaystyle \sqrt{-i}=\{\cos(270+k.360)+i\sin(270+k.360)\}^{1/2}\\=\cos(135+k.180)+i\sin(135+k.180)$

$\displaystyle k=0: \; =\cos(135)+i\sin(135)= -\frac{1}{\sqrt{2}}+i\frac{1}{\sqrt{2}},$

$\displaystyle k=1:\;=\cos(315)+i\sin(315)= \frac{1}{\sqrt{2}}-i\frac{1}{\sqrt{2}}$.

Other integer values for k simply repeat these, so just two possibles.

Tiggsy

find the equation for predicting y from x

Your  question is not complete.

What would you like to sketch?

sin 90degrees = 1 

I know this becasue sin(angle) is given by the y value on the unit circle.

You make a log by cutting down a tree.   

Unless it is an icecream log. Then you would need a freezer and and some cream. 

Thanks guest that makes sense.

I'll just write my version of what you have said in LaTex.

$\begin{align} \sqrt{-i}&=\sqrt{\frac{1-2i+1}{2}}\\ &=\sqrt{\frac{1-2i-i^2}{2}}\\ &=\sqrt{\frac{(1-i)^2}{2}}\\ &=\frac{\sqrt{(1-i)^2}}{\sqrt 2}\\ &=\frac{1-i}{\sqrt 2}\\ &=\frac{\sqrt 2(1-i)}{2}\\ &=\frac{\sqrt 2}{2}-\frac{\sqrt 2\;i}{2}\\ \end{align}$

Ok that is good although it is only one of the 2 answers

With the aid of a quick sketch I can see that the 2 roots are

$\sqrt{-i}=\pm \left[\frac{\sqrt{ 2}}{2} (1-i)\right]$       

I still cannot see the error in my #2 answer though    

I’d like to know how someone that smart can also be that pretty .

How do we know your not her? If you werent cool with it then you would deny it. I don’t know why though because you are really beautiful.  If I make an account on here will you talk to me?

I swear I wont be rude.

No, I said it is not me.

Oh it is you then. You are very pretty hectictar. 

Here it is :)