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$$$$f(x)=(5x-4)^\frac{1}{2}$$
$$f'(x)=\frac{1}{2}(5x-4)^{-\frac{1}{2}}*5=\frac{5}{2(5x-4)^{\frac{1}{2}}}$$
$$f'(4)=\frac{5}{2(5*4-4)^{\frac{1}{2}}}=\frac{5}{2(20-4)^\frac{1}{2}}$$
$$=\frac{5}{2*\sqrt{16}}=\frac{5}{2*4}=\frac{5}{8}$$

The chance is 5*10⁶/7*10⁹ or 5/7000.  That is, there are 5 chances in 7000 of being born in that nation (or 1 in 1400 if you prefer).  

Just type sin(63) into the calculator here and press = or Enter

If you've two sides and the included angle use the Cosine rule.

If you've three sides use the Cosine rule.

If you've two angles and a side use the Sine rule.

In the case when there is an even number of numbers it is normal to take the median as the average of the two nearest the middle.  So, in this case the median would be (33+43)/2 = 38

model nucleus/model atom = actual nucleus/actual atom

n/5 = 10^-15/10^-10

n/5 = 10^-15*10⁺¹⁰ = 10^-15+10 = 10^-5

Multiply both sides by 5

n = 5*10^-5 cm

n = 0.00005 cm!  She needs a bigger model!

You can simplify it as follows (where the capital pi symbol means take the product of what follows).  However, the site calculator doesn't have this product function as far as I can tell.

$$\begin{tabular}{|c|c|c|c|c|c|}
\hline
$x$&1&8&10&-20&25\\\hline
$y$&-6&15&21&-69&66\\\hline
\end{tabular}$$

Now I notice that the bottom numbers appear to be a bit less than 3 times the top ones.

so I am going to experiment with that;

$$\begin{tabular}{|c|c|c|c|c|c|}
\hline
$x$&1&8&10&-20&25\\\hline
$3x$&3&24&30&-60&75\\\hline
$y$&-6&15&21&-69&66\\\hline
\end{tabular}$$

Now I notice that 3x-9=y

So the rule is

$$y=3x-9$$

$$I(t)=I_0 e^{-\lambda t}$$

well...

if you go ahead and graph these points they appear to all be on a line.

So take a couple of them and figure out what the slope and intercept is and then double check that they all do lie on that line.

I get a slope of 3 and an intercept of -9 so the line the points lie on is

y=3x - 9

Checking this we get

1 -> -6

8 -> 15

10 -> 21

-20 -> -69

25 -> 66

and you can see that all the points lie on this line.

Note that .10 = 10%

And .01 = 1%

So it appears that the "rule" is........move the decimal point two places to the right and add a percent sign...

Can you do these two, now???

if you mean what per cent of 330 is 295.5

$${\frac{{\mathtt{295.5}}}{{\mathtt{330}}}}{\mathtt{\,\times\,}}{\mathtt{100}} = {\frac{{\mathtt{985}}}{{\mathtt{11}}}} = {\mathtt{89.545\: \!454\: \!545\: \!454\: \!545\: \!5}}$$

so 295 is 89.55% of 330

if you mean what percent of 295.5 is 330

$${\frac{{\mathtt{330}}}{{\mathtt{295.5}}}}{\mathtt{\,\times\,}}{\mathtt{100}} = {\mathtt{111.675\: \!126\: \!903\: \!553\: \!299\: \!5}}$$

so 330 is 111.68% of 295.5

Calculate AB

Calculate the length of arc AB

Calculate the area of the shaded region 

However I do not know how, and any help is appreciated, please and thank you!

I couldn't resist putting you picture in here.  It is easy done with Desmos Graphing calculator.

Plus, "How to upload an Image"  is one of the sticky topics, in the bottom right of the page.

Are you happy with your answer?

3-th square of (-125):

$$\\\sqrt[3]{-125}=\sqrt[3]{125}\times\sqrt[3]{-1}=(5)* \left\{ e^{i\frac{1}{3}(\pi+0*\pi)},
e^{i\frac{1}{3}(\pi+2*\pi)},
e^{i\frac{1}{3}(\pi+4*\pi)}
\right\}\\
\sqrt[3]{-125}=(5)* \left\{
e^{i\frac{1}{3}\pi},
e^{i\frac{3}{3}\pi},
e^{i\frac{5}{3}\pi}
\right\}\\
\sqrt[3]{-125}=(5)* \left\{
e^{i\frac{1}{3}\pi},
e^{i\pi},
e^{i\frac{5}{3}\pi}
\right\}\\
\boxed{e^{i\pi}=-1}\\
\sqrt[3]{-125}=(5)* \left\{
e^{i\frac{1}{3}\pi},
-1,
e^{i\frac{5}{3}\pi}
\right\}\\ \\
\sqrt[3]{-125}=(5)* e^{i\frac{1}{3}\pi} \quad \text{complex number} \\\\
\sqrt[3]{-125}=(5)*(-1)=-5\quad \text{real number }\\\\
\sqrt[3]{-125}=(5)* e^{i\frac{5}{3}\pi} \quad \text{complex number} \\$$

Hi admin, you made a small mistake

Difference means subtract.

Let x be the little one and y be the big one

y-x=25 (1)

y/6=x ==> y=6x (2)

sub (2) into (1)

6x-x=25

5x=25

x=5

y=6*5=30

584/25

I think it is better written as a mixed numberal

$$\frac{584}{25}=23\frac{9}{25}$$

Alan already worked out that the gradient was 3

y-4=3(x-2)

just rearange that to get the equation in the form that you want.

Yes It was good reasoning by GoldenLeaf 

I would like to set it out a little differently though.

$$\frac{\sqrt{32}}{2}<\frac{\sqrt{36}}{2}\\\\
\mbox{therefore}\\\\
\frac{\sqrt{32}}{2}<\frac{6}{2}\\\\
\frac{\sqrt{32}}{2}<3\\\\
and\\\\
\frac{\sqrt{32}}{2}<3<4\\\\
therefore\\\\
\frac{\sqrt{32}}{2}<4\\\\$$

So yes, 4 is not greater than sqrt(32)/2

Excellent logic GoldenLeaf

The "Julian Date" of 1.2.2014 0:0:0 o'clock is 2456689.50000

The "Julian Date" of 1.2.2014 12:0:0 is 2456690.00000

The "Julian Date"

see:

http://aa.usno.navy.mil/data/docs/JulianDate.php

The Julian date for CE 2014 February 1 00:00:00.0 UT is JD 2456689.500000

The Julian date for CE 2014 February 1 12:00:00.0 UT is JD 2456690.000000 

The Julian date for CE 2014 May 29 12:00:00.0 UT is JD 2456807.000000 

Let me just add to what reinout said....if we took the second derivative of the first function, we get "2," which indicates that the curve is concave upward, and the second derivative of the second function = -2, which indicates the curve is concave downward.

And as Melody indicated, both parabolas share the same vertex - (0,1).

Compared to what....The Great Pyramid ????

Let the number of litres of 30% concentration = x

And since we have 120 litres as a final amount, then the number of litres of the 60% solution is just

(120-x).    So we have

.30x + .60(120-x) = .40(120)

.30x + 72 - 60x = 48   Subtract 72 from both sides and simplify the left side

-.30x  = -24        

.30x = 24      Divide both sides by .30

x = 80  And that's the number of litres of the 30% solution we need

And (120 - x) = (120 - 80) = 40 And that's the number of litres of the 60% solution we need

You will find it in Formulary at the top of the page!  Look there.

In statistics range refers to the highest score - the lowest score

So the range is    $${\mathtt{10\,000}}{\mathtt{\,-\,}}{\mathtt{1\,273}} = {\mathtt{8\,727}}$$

Reinout is right ofcourse - but my goodness, what a lot of explaination!  

It's simple. 

The parabola just gets flipped upside down!