MaxWong

Because F = ma, if the acceleration is 0, then the forces are in equilibrium.

First, the man have weight(of course!), so there is a force pointing downwards labeled W(stands for weight).

And then, there is tension in the string, so there is a force pointing upwards labeled T(stands for tension).

Also, there is normal force, so there is another force pointing upwards labeled N(stands for normal force).

Finally, the normal force and tension balances the weight, so it should be in equilibrium.

P.S.: I am not specifically good at Physics, but I tried my best to help. If anyone sees mistake in my solution, please teach me. Thank you.

"The interior acute angles are congruent" tells you it is a regular pentagram, so the measure of interior acute angle = 180/5 = 36 degrees. (I divided by 5 because it is a pentagram(5-gram) .)

Now you draw a straight line like this:

Calculate the angles in the triangle formed:

the base angles of the isosceles triangle is 72 degrees.

As the line drawn is a straight line, so the exterior angle is 108 degrees.

Measure of internal reflex angle = 72 * 2 + 108 degrees = 252 degrees

You see, the first one is a hexagon(6-gon), so the sum of interior angle is: 

\((6-2)\cdot(180^{\circ}) = 720^{\circ}\)

The second one is a pentagon(5-gon) although it is a concave pentagon. But in fact, you still use the same formula for this.

The sum of interior angle is:

\((5-2)\cdot(180^{\circ}) = 540^{\circ}\)

Edit:

Explanation for (n-2)*(180 degrees):

For any polygon, start from one of the vertices, then repeatedly draw an edge to another vertice, until all the vertices are directly connected with an edge. If the polygon is an n-gon, then it should now be divided into (n-2) triangles. So the sum of  interior angles should be (n-2)*(180 degrees).

Notice: \(\dfrac{n(2n+1)(7n+1)}{6} = (n+1)^2+(n+2)^2+\dots+(2n)^2\)

So the question is equivalent to: Find all integers n such that \((n+1)^2+(n+2)^2+\dots+(2n)^2\) is an even number.

But if x^2 is an odd number, then x is an odd number, also if x^2 is an even number, then x is an even number(why?)

So the question is just asking: Find all integers n such that \((n+1)+(n+2)+\dots+(2n)\) is an even number.

Next, we will find the formula for \((n+1)+(n+2)+\dots+(2n)\).

\(\quad(n+1)+(n+2)+\dots+(2n)\\ =n^2+(1+2+3+\dots+n)\\ =n^2+\dfrac{1}{2}(n)(n+1) \\ = \dfrac{n}{2}(3n+1)\)

So, finally, the question is equivalent to: Find all integers n such that n(3n+1) is divisble by 4.

For some reason, we arrived at a simpler question. Let's try to solve it now.

n = 4k for k = 1,2,3,4,... is a trivial solution, because if n is divisible by 4, then n(3n+1) is also divisible by 4.

n = 4k+1 for k = 0,1,2,3,4,... is another set of solution. Let's substitute n = 4k+1 into n(3n+1) to see why.

n(3n+1) = (4k+1)(12k+3+1) = (4k+1)(12k+4) = 4(4k+1)(3k+1)

Therefore, when n = 4k+1, for k = 0,1,2,3,4,..., it is also a solution.

Final solution: n = 4k for \(k\in \mathbb Z^+\), or n = 4k+1 for \(k\in \mathbb Z^*\)

First we need to find how far it moves in an hour.

1 year = 365 1/4 days = 8766 hours

So the plate moves 1/5844 inches/hour

Multiplying by 14: The plate moves 7/2922 inches in 14 hours.

Part(a):

\(\quad a+(a+1)+(a+2)+\dots+(a+n-1)\\ =(1+2+3+4+5+...+(a+n-1)) - (1+2+3+4+5+...+a-1)\\ \boxed{\text{Know: }1+2+3+4+...+ n=\dfrac{n(n+1)}{2}}\\ =\dfrac{(a+n-1)((a+n-1)+1)}{2} - \dfrac{(a-1)((a-1)+1)}{2}\\ =\dfrac{(a+n)(a+n-1)}{2}-\dfrac{a(a-1)}{2}\\ =\dfrac{a^2+2an+n^2-a-n}{2}-\dfrac{a^2-a}{2}\\ =\dfrac{2an+n^2-n}{2}\\ =an+\dfrac{n(n-1)}{2}\)

Put it on a coordinate plane.

Now A(0,6), B(0,0), C(\(3\sqrt3\),0)

Slope of AM = \(\dfrac{0-6}{\dfrac{3\sqrt3}{2}-0}\)=\(-\dfrac{4\sqrt3}{3}\)

Equation of AM is \(y = -\dfrac{4\sqrt3x}{3}+6\)

Slope of CN = \(\dfrac{0-3}{3\sqrt3-0}\)=\(-\dfrac{\sqrt3}{3}\)

Equation of CN is \(y=-\dfrac{\sqrt3x}{3}+3\)

Solving for the intersection point(P):

P = (\(\sqrt3\),2)

Recalling that C = (\(3\sqrt3\),0).

Length of CP = \(\sqrt{(\sqrt3-0)^2+(3\sqrt3-2)^2}\) = \(\sqrt{34-12\sqrt3}\).

And this is the simplest it can get.

Yes it helps. Thank you :D

27^1986 = 48801775164233180193152186583304417336797680575180581115895393460515804997481426460328920628366330784510548613573893133597581628998231738095941136304757747700966133994279657641604503248144993063886307509474264646012144202451717134988223435658134723146939548526526046746149774508324183700407959892199488896439369356604901578709343457074224182222533446701001327664448924429135151145101780007678647192526652283864386426058549923535427662270800539978182246623630215550544972216345285441276619530080398735910161756589449840397939641361894901562665980047466740082396373341778906897800555115722452770664688076387259535876436046972899204638534393769005219725810593392077925096955164314720390198579103842847954992632961329117315094216615129428536324654588082306372021647022430577611704382369124740553707849232982840968333108506219759397531460279020717574204357008717885397405312628389031757241409912778322513652225905208643086567820940013264513686836151697257293488310351014146560735095573779572404778048653597239139290808021905056342235736193779153753316842586904178196600043336256641895973206688460099090120406045255624825627683977323696929363287506689433912930661175286684677160661203889391570143445392987720450701894381239294874928803004177216894164745421763945685814401999598340303489036530867374232572727166897181905438211435777103860575566122738459632273930151354490629643457508872730539666645848055176086262500173450757837780931124099511147443471470828134586813514044949515967779408501414682996812976558402552553142807980868977183835493023048436656516057263092336591331513716603390064194308624685089075088054532483604955744721552650388426853236066590978184087032928185890064270903768451944629764513609533788042180992515318842081697688028887938906492323565305085808157568546090292860937463322677394902590580918646428717626492549592383518428731724982544318398065913196769349927074541649454968742704298225424732774113924319005257485049325235138238093730273646224858897535235298885105594129943937351459724702482583667236536502693815709185789808126712796159398263200073561808422405408703688494449817834260263098757025774244181566762357315704174052124364390755529262395531978098406249555755014544306699077239975625672904487308441554678704578185924139361375521658379381070765883331271855520007826465075200564387537336981534406153890434416074062845454707273089136109080238153308852346576731949235793839884975531153052962588362467246851989966711345919740451341621252450778752838477266614739871237844364433820224901541153831413862665959090015057100579979531958425068248597012156787283346913344245335741283554176163827879328047740051313782996843172666708463050608416638964963699210219313787074584062514104031192818402598712368534388109017171715451967516936812146050223558204491606779630732845610748256709841526809731630661112279428213058346512493428425689

27^1986 = 89 mod 128

Therefore last 7 digits = 1011001.

I calculated it using a program, so... :P

Let r(p) be the revenue.

r(p) = p(128-4p) = -4p^2 + 128p

Completing the square, r(p) = -4(p^2-32p) = -4(p-16)^2 + 1024.

Therefore, if the price is 16 dollars, the revenue will be maximized, which will be 1024 dollars.

Assuming you mean f(0), f(-2),f(3) instead of F being the antiderivative of f.

f(0) = 3(0) - 4 = -4

f(-2) = 3(-2) - 4 = -10

f(3) = 3(3) - 4 = 5

Now assume you mean the antiderivative.

F(0) = 3(0)^2 / 2 - 4(0) = 0

F(-2) = 3(-2)^2 / 2 - 4(-2) = 14

F(3) = 3(3)^2 / 2 - 4(3) = 3/2