Melody

What is a radian and what is the plural form of radius cause I was watching khanacademy and he said radiuseseseses so I'm confused.

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For me the plural of radius is radii  BUT  my country (Australia) uses the same versoin of English as England does.

The USA has changed much of the spelling to make it easier. So maybe for them the plural of radius is radiuses.

A USA person will need to comment on this.

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The words radius and radian are closely related.

The circumference of a circle is  \(2\pi r\)

And there are  \(2\pi\)  radians in a revolution.

so                      \(\boxed{2\pi\;\;radians\;\;=\;\;360^\circ}\)

Rearranging this equations we get.

\(2\pi\;\;radians\;\;=\;\;360^\circ\\ 1\,radian=(\frac{180}{\pi})^\circ\\ or\\ 1^\circ=\frac{\pi}{180}\;radians\\ \)

Say a  number is       6.126543

Then 6 is the floor value

OR

6 is the number rounded down to the nearest integer. 

Just so you will know for next time

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A question for someone else.  Maybe Alan 

What is the floor value of       -6.5                   [Or any number between -7 and -6]

I assume it is -7, is that correct ?

Just put it in exacly as Alan has written it. Replace A with the Area you ahve been given.

Exactly what buttons to push depends on what calc you are using.

\(s=\sqrt{4\frac{A}{5}\tan{\frac{\pi}{5}}}\)

A= 387

sqrt(4*387/5*tan(pi/5)) = 14.9979187446241353 = 15

which is the same I got below. 

what is the perimeter of a regular pentagon whose area is 387cm2

\(Area \;of \;triangle=0.5abSinC\\ 77.4=0.5r^2Sin72\\ \frac{77.4}{0.5Sin72}=r^2\\ r=\sqrt{\frac{77.4}{0.5Sin72}}\)

r= sqr(77.4/(0.5*sin(72))) = 12.757991703716932

\(L^2=a^2+b^2-2abcosC\\ L^2=2r^2-2r^2*cos 72\\ L^2=a^2+b^2-2abcosC\\ L^2=2r^2-2r^2cos 72\\ L^2=2*12.757991703716932^2(1-cos(72))\)

sqrt(2*12.757991703716932^2(1-cos(72))) = 14.997918744628501

5*14.997918744628501 = 74.989593723142505

I get the perimeter to be 75cm long

What do you mean by unit value and decimal value?

If the unit value of the square root of 17 is represented as x and its (sq root 17) decimal value is represented by y,  x^2+(1/y)?

I will try and understand what you have said first.

\(x=\sqrt{17} = 17^{0.5}\\ y=\sqrt{\sqrt{17}}=17^{0.25}\\ x^2+\frac{1}{y}\\=17+\frac{1}{17^{0.25}}\\\approx 17+\frac{1}{2.0305431848689307}\\\approx17.4924790605054523308\)

P(0,1)       Q(x)=e^x

Let the  gradient of the secant PQ = m

 \(m=\frac{Q(x)-1}{x-0}=\frac{Q(x)-1}{x}\\ When \;\;x=-0.1\qquad\;\; m=\frac{e^{-0.1}-1}{-0.1}=0.952\\ When \;\;x=-0.01\qquad m=\frac{e^{-0.01}-1}{-0.01}=0.995\\ When \;\;x=-0.001\qquad m=\frac{e^{-0.001}-1}{-0.001}=1.000\\ When \;\;x=0.001\qquad m=\frac{e^{0.001}-1}{0.001}=1.001\\ When \;\;x=0.01\qquad m=\frac{e^{0.01}-1}{0.01}=1.005\\ When \;\;x=0.1\qquad m=\frac{e^{0.1}-1}{0.1}=1.052\\\)

Just looking at thes numbers it seems reasonable that the gradient of the tangent to y=e^x where x=0 is about 1

(And in fact it really is 1)

What is the shape?

Jeff, why would I bother to help you when you are too rude to respond to my answers?

See here (just for an example)

Find the sum of the real values of x such that the infinite geometric series \(x+\frac{1}{2}x^3+\frac{1}{4}x^5+\frac{1}{8}x^7+\dots\text{ is equal to }-12\)

This is the sum of a GP

\(a=x\\ r=0.5x^2\\ S_\infty=\frac{a}{1-r}\;\;where\;\;|r|<1\\ |0.5x^2|<1\\ 0.5x^2<1\\ x^2<2\\ -\sqrt2

\(S_\infty=\frac{x}{1-0.5x^2}\\ -12=\frac{x}{1-0.5x^2}\\ -12(1-0.5x^2)=x \\ -12+6x^2=x \\ 6x^2-x -12=0 \\ 6x^2-9x+8x -12=0 \\ 3x(2x-3)+4(2x-3)=0\\ (3x+4)(2x-3)=0\\ x=-4/3\quad or \quad x=3/2\\ 3/2 >\sqrt2\quad \text{so it is illiminated}\\~\\ \therefore\;\;x=-1\frac{1}{3} \)

check

\(a=-4/3\quad  \\  r=0.5*16/9 = 8/9\\ S_\infty=\frac{-4}{3(1-\frac{8}{9})}\\ S_\infty=-12\\\)