# Posts by STEVE

Total # Posts: 51,196

Economics
n?xy - (?x)(?y) ----------------------------------------------- ?(n?x^2 - (?x)^2) ?(n?y^2-(?y)^2)

Economics
n?xy - (?x)(?y) ------------------------------------------- ?(n?x^2 - (?x)^2) ?(n?y^2-(?y)^2)

Economics
n?xy - (?x)(?y) --------------------------------- ?(n?x^2 - (?x)^2) ?(n?y^2-(?y)^2)

Economics
?(x-x?)(y-y?)

Calculus II: Shell Method for finding volumes
I guess the decimal values are ok, but I tend to prefer the exact values of (-1+?17)/2 and (-1-?17)/2. They are a bit cumbersome to work with by hand, but with the wealth of online web sites to do the calculations, that should be no obstacle. Your limits of integration are ...

calculus
starting with the peak at t=0 suggests use of a cosine function. Since the period is 11 years, start with y = cos(2?/11 t) The amplitude is (120-0)/2 = 60, so y = 60 cos(2?/11 t) The center line is at y=60, so y = 60 cos(2?/11 t) + 60 where t is the number of years since 2000.

Algebra
a^x is growth if a > 1 a^x is decay if a < 1 a^x passes through (0,1) for any value of a.

math
x + x+1 + x+2 = 3x+3

math
If you say so.

Math
see related questions below

Math
1/6 * 3/2 = 1/4

Math
It might help to express them all to the same precision: 0.125 0.100 0.120 ok?

Science
Take a look at Snell's Law. Then you can make up measurements and figure the related values. Not sure how you'd lay a protractor against the water container... I'm sure you can find some online examples, though.

Physics
Using the usual trig angles (not compass headings), 15 cis45° + 18cis5° = 28.54+12.18i = 31.03 cis23.11°

calculus
the initial value of 11 means that 2sin(2pi/3(t+1/4)) = 1 sin(2pi/3(t+1/4)) = 1/2 2pi/3(t+1/4) = pi/6 t + 1/4 = 1/4 t=0 With no displacement, y(0) = 0 since sin(x) is a rising curve at x=0, the graph needs to be shifted left to bring the value of 1 into play. Try playing ...

Math
well, 3577.4 is less than 3887, so ... how many feet in how many years?

you're kidding, right? It has to be right below A and to the left of C. Where do those lines cross? Or, see that point D has to have the same x-value as A, and the same y-value as C.

Algebra
x^2 + 8x + 16 = 10+16 (x+4)^2 = 26 x+4 = ±?26 x = -4 ±?26 Nope, not A

Math
Since you don't say that each 5m portion is a separate attempt (like walking up stairs), I'll assume that his slippage is continuous as he climbs. That is, it takes him 5*5=25 seconds to climb 5-1=4 meters. So, his rate of ascent is 4/25 m/s It will take him 53/(4/25...

trigonometry
In QI, if cosx = 4/5, sinx = 3/5 (1+tanx)/(1-tanx) = (cosx+sinx)/(cosx-sinx) = (4/5 + 3/5)/(4/5 - 3/5) = (7/5)/(1/5) = 7

impatient much?

Maths

Maths
Huh? You already said that |AB|=4cm To construct the figure, bisect a 60° angle to get angle A. Mark B where an arc of radius 4 intersects one side of angle A. C lies on a circle CB of radius 5 with center at B. D lies on a circle CA of radius 5 with center at A. So, CD is...

Maths
surely you can plug in your numbers s = r? 60° = pi/3 (not pie!!)

Deduction
P => Q says nothing about what happens if ~P.

Math
a/m = 3/7 a+92 = m+40

math
statement: for all x, either p(x) or q(x) There is some x where p(x) and q(x) are both false. The mod works just like any other function x ?3 1 Since x is congruent to 3 (mod 1) makes no sense, you must mean x ? 1 (mod 3) We all know what that means, so just negating the ...

Pre-Calc
I assume you mean r = 21/(5-2cos?) When you consider the general equation for a conic: r = ep/(1 - e cos?) you see that you must set it up as r = 21/[5(1 - 2/5 cos?)) Now you can read off e and p. Since e<1, you have an ellipse. Now you need to use the distance to the ...

Maths
since cos2x = cos^2x - sin^2x, you have 1 - 2cos2x = sin2x 1 - 4cos2x + 4cos^2 2x = sin^2 2x 5cos^2 2x - 4cos2x = 0 cos(2x)(5cos(2x)-4) = 0 cos 2x = 0 cos 2x = 4/5

substitution and elimination methods
x-4y = 20 -3x+4y = -12 add them together and you get -2x = 8 x = -4 Now use that to get y.

Math
Just divide the figure up into rectangles. You can then see the width and height of each part, and add up all the smaller areas.

Math
yep

Math
then the base is 24 and the height is 8 Now you can figure the area.

MATHS
You want a power of two to be x^2+4x+4 = (x+2)^2 So, which powers of 2 are perfect squares? There are lots of answers: (6+2)^2 = 64 = 6^2 + 4*6 + 4 = 1446 = 2^6 (14+2)^2 = 256 = 14414 = 2^8 All even powers of 2 are perfect squares.

Physics
m = 691/9.8 (c) F = ma

maths
This is exponential decay. No is the starting value e^-k is the decay rate N(t) is the amount remaining after t time intervals. Surely your text laid all this out in the section on exponential growth/decay.

Calculus
Hmmm. I found a proof in about 15 seconds. First, you need to understand what it means to be integrable: If a function is continuous on a given interval, it’s integrable on that interval. Now, if you show that f(x) is discontinuous at any point in [0,1] then it is not ...

math
clearly d = -k/3 S30 = 30/2 (2k + 29(-k/2)) = -375k/2

go with @Reiny
I calculated the difference, then erroneously assigned that to x. My bad.

Trigonometry
(25 - 1/2)/2 = 49/4 so, x = 49/4 (32 1/2)/(50/4) = y/(32 1/2) y = 169/2

maths
(x+10)/x = 3/1 x = 5 So the numbers are 5:15:20

Math 8
What do you mean "solve it separately" ? It's a system. If you rearrange things to match, you have 2x-y = 4 2x-y = -4 Clearly there are no solutions; the lines are parallel, and do not intersect.

math
that would be 1.20^3 = 1.728 or 72.8% increase

calculus
they are the same curves, just shifted. It usually depends on whether the initial value is a maximum or a center-value.

Maths
Just plug and chug. You know that f = sin f' = cos f" = -sin f(3) = -cos So, f(x) = f(?/3) + f'(?/3)(x-?/3) + f"(?/3)/2! (x-?/3)^2 + f(3)(?/3)/3! (x-?/3)^3 + ... f(x) = ?3/2 + (1/2)(x-?/3) - (?3/2)/2! (x-?/3)^2 - (1/2)/3! (x-?/3)^3 + ... f(x) = ?3/2 + (1/2)(x...

Math
3:1:4 = 9:3:12

calculus
If you mean ?[-3,3] x^2+2x+1 dx then the integral is x^3/3 + x^2 + x [-3,3] = (9+9+3)-(-9+9-3) = 24 or, ?(x+1)^2 dx = 1/3 (x+1)^3 [-3,3] = 1/3 (4^3+2^3) = 24

maths
just pick any element, say x. The the subset with that one element is {x}

Calculus
a simple google on the question turns up many discussions of the problem and its proof.

Math (Integrals)
You have the right idea, but you are working a definite integral, so there is no constant C. Instead, evaluate A(t) at the interval endpoints: A(x) = (4t + 4/t)[1,x] = (4x + 4/x)-(4*1 + 4/1) = 4x + 4/x - 8

Math - Calculus
It's just the chain rule. If F(t) = ?f(t) dt then ?[v(x),u(x)] f(t) dt = F(u)-F(v) so, taking derivatives, that gives you f(u) u' - f(v) v'

Math
j/p = 4/7 (j - j/2)/(p-60) = 1/2 (j/2)/(7j/4 - 60) = 1/2 4j = 7j-240 j = 80

Pre Algebra
In 1 hour he can bake 4 times as many. So, what is 4 times 2 1/2?

Math (Definite Integrals)
Do not assume r=1. But, it is a constant. And you are right, the graph is a sem-circle, so the area is ?/2 r^2 Not sure where the a comes in, but to do the integral, now is the time to start getting used to trig substitutions. Let x = r sin? r^2-x^2 = r^2(1-sin^2?) = r^2 cos^2...

C++
I assume you have picked up some C++ by now, so you know how to build an object and define variables. The logic can be something like what follows, but expressed in C++, using cout and cin print "enter mass: " read mass weight = mass * 9.8 print "Weight: ",...

Algebra 2
100(1 + .03/4)^(4*20) = ?

Pre-Cal: Limacons
you might want to play around on wolframalpha.com by varying the parameters. You have the equations for sample curves, as well as examples in your book and online. You know the general form is r = b + a cos?. And of course, since sin? = cos(?/2 - ?) the two are just rotations ...

Eigenvalues
using your standard methods, see whether you come up with this: http://www.wolframalpha.com/input/?i=eigenvalues+%7B%7B10,10%7D,%7B-4,-3%7D%7D

Mathematical Induction
prove that P(1) is true: 8 = 1/2 1(3*1+13) = 16/2 = 8 Assuming P(k), see what P(k+1) means: 8+11+...+(3k+5)+(3(k+1)+5) = k/2 (3k+13) + (3(k+1)+5) = k/2 (3k+13) + 3k+8 1/2 (3k^2+13k + 6k+16) = 1/2 (3k^2+19k+16) = 1/2 (k+1)(3k+16) = 1/2 (k+1)(3(k+1)+13) = P(k+1) So, P(1) and P(k...

Calculus
f"(x) = 2+cos(x) f'(x) = 2x + sin(x) + c1 f(x) = x^2 - cos(x) + c1*x + c2 f(0) = -1+c2 = 5 --> c1=6 f(?/2) = ?^2/4 + c1*?/2 + 6 = -3 f(x) = x^2 - cos(x) + (?/2 - 18/?)x + 6

calculus
To evaluate ?e^(-?x) dx let z^2 = x 2z dz = dx and now you have ?e^-z 2z dz Now let u = z dv = e^-z dz du = dz v = -e^-z dz ?u dv = uv - ?v du, so ?2z e^-z dz = -2ze^-z + 2?e^-z dz = -2ze^-z - 2e^-z Now evaluate that at the limits and you get ?[0,?] e^(-?x) dx = 2

calculus
To evaluate ?x/(x^4+25) dx let z^2 = x 2z dz = dx now you have 1/2 ?1/(z^2+25) dz That is just a standard integral, giving (1/2)(1/5)arctan(z/5) evaluate at the limits and you get -?/20

calculus
let u=x^2 and it's a cinch.

Algebra
3/5 + 2/x = 1 now just solve for x

Math
P(red) = (#red)/(#total) Now take 90*P(red)

calculus
let u^2 = x-1 x = 1+u^2 2u du = dx Now your integrand is 1/(1+u^2) * u * 2u du = 2u^2/(1+u^2) du = 2(1 - 1/(1+u^2)) du That integrates to 2(u - arctan(u))+C = 2(?(x-1) - arctan?(x-1))+C

calculus
using partial fractions, you have 11 * 1/[(x-5)(x^2+5x+25)] = 11/75 * [1/(x-5) - (x+10)/(x^2+5x+25)] Doesn't look much better, does it? Well, the 1/(x-5) integrates easily enough. The other term has to be worked into something more standard. x^2+5x+25 = (x + 5/2)^2 + 75/4 ...

Calculus
Using a scaled-up 3-4-5 triangle, the distance after 10 minutes is 1/6 (500) miles. If z is the distance between them after t hours, then z^2 = (300t)^2 + (400t)^2 z dz/dt = 2*300^2 t + 2*400^2 t Now just plug in your numbers with t = 1/6

calculus
Using partial fractions, (x^2-x+12)/(x^3+3x) = 4/x - (3x+1)/(x^2+3) The 4/x integrates easily enough. The rest has to be worked on... 3x/(x^2+3) is just another log. For the rest let x = ?3 tan? x^2+3 = 3 sec^2? dx = ?3 sec^2? d? 1/(x^2+3) dx = 1/?3 d? Now, putting all that ...

Algebra
I'll assume the usual carelessness with parentheses and spacing, and go with (a+5/(4a)) + 11/12 = 2/(3a) Using a common denominator of 12a, that gives a(12a) + 3*5 + 11/12 (12a) = 2*4 12a^2 + 15 + 11a = 8 12a^2+11a+7 = 0 That has no real solutions. So, if I interpreted you...

math
all multiples of 3 and 7, added together: 3,6,9,12,... 7,14,21,28,... and sums of those rows in all combinations.

calculus
just use the chain rule twice y = v^2(u(x)) y' = 2v v'(u) u'(x) y' = -2sin(1/x) cos(1/x) (-1/x^2)

Math calculus
v = ?r^2h = 5, so h = 5/(?r^2) a = 2?r^2 + 2?rh = 2?r^2 + 10/r da/dr = 4?r - 10/r^2 da/dr=0 when 4?r^3=10 r = ?(5/(2?)) h = 5/(?r^2) = 5/(?(5/(2?))^(2/3)) = ?(20/?) since 3 > 2r > h, it works.

Math bet
(1.2 * 10^-4 km)(6.25*10^-5 km) = 7.5*10^-9 km^2 that's 7.5*10^-3 m^2

geometry/check my work
D it is When you combine translations all bets are off.

physics
recall that the standard equation of motion means that you will solve for h in h(t) = h + vt - 4.9 t^2 = 0 Just plug in your initial speed v and time = 30, then solve for h.

English
They are all used. #1 is most common in my experience. Take that -- I've had it! Get it?

Math
divide total miles by total days

Math
I'd say 20/4 = 5 Of course, if he didn't give away all of the quarters, that could change things.

further algebra
The nth bounce will have height 30 * (4/5)^n No idea how many bounces it will take before stopping. But, to get the total travel after n bounces, just add 30 to twice the sum of n terms of the sequence (a round trip is up and back down)

Precalculus
Multiply top and bottom by 1+sin?: 1/(1-sin?) * (1+sin?)/(1+sin?) = (1+sin?)/(1-sin^2?) = (1+sin?)/cos^2?

Math
it has as much symmetry as a regular octagon. Rotate it 1/8 of a circle and it looks the same as before. (As long as the cars are all the same color!)

maths
a+2d = a + a+d 5/2 (2a+4d) = 30 Now just crank it out.

Calculus
Review the chain rule d/dx f(u(x)) = df/du * du/dx

Math
(x,y) -> (x+1,y-1) so, move right 1 and down 1 It usually helps to plot the points on some graph paper, eh?

Physics
You should subtract the 5 meters from the calculated distance, or the package will land 5 meters from shore in the water, rather than on land! 2772 - 5 = 2767 You added an extra 5 m/s to the speed!

Physics
how long does it take to fall 125 meters? 4.9t^2 = 125 Using that value of t, how far will the package move at 550 m/s ? You might want to consider an extra 5 meters to hit at the center of the island?

Physics
clearly he wants the vertex of the parabola to be just above the wall. Recall that the trajectory can be described as y = x tan? - g/[2 v^2 cos^2?] x^2 Plugging in your values, that is y = 0.707x - 0.196x^2 The vertex is at (1.8,0.6) So, a distance of 1.8 meters will provide ...

Physics
good work

Physics
surely by now you know that distance = speed * time

Physics
how long does it take to fall 10 meters? 4.9t^2 = 10 The horizontal speed stays constant, so using the value of t calculated above, how far does he go? If less than 5.5 meters, he's gonna break something...

PreCalculus
I'd say that it would be 0.28 * 70000/12

Math
zero

Calculus
I'll do one, and you can follow the method for the others. g"(x) = 12x^2 g'(x) = 4x^3+c g'(1)=7, so 4+c=7, so c=3 g'(x) = 4x^3+3 g(x) = x^4+3x+c g(1)=3, so 1+3+c=3, so c=-1 g(x) = x^4+3x-1

math
x+10 = 4(x-5)

Calculus - Riemann Sum help
You just have to figure out how many intervals you want to use (n). Then, as it says, the interval width is ?x=2?/n The interval is apparently [0,2?]

Math
5a/6 - a/3 - 2/3 = -1/6 a(5/6 - 1/3) = -1/6 + 2/3 a(1/2) = 1/2 a = 1

algebra
n/s = 5/3 n-25 = s+25 5s/3 - 25 = s+25 2s/3 = 50 s=75 so, n=125 n+s=200

Math - Calculus
81 ? (2i-1) i=1 = 2?i - ?1 = 2*81*82/2 - 81 = 6561 since you know that n ?i = n(n+1)/2 i=1 Or, working with the sum as an AP, you have a=1 d=2 S81 = 81/2 (2+80*2) = 6561 Or, note that you have the sum of the first 81 odd numbers. That sum is 81^2 = 6561

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