Wednesday
April 16, 2014

Posts by Reiny


Total # Posts: 25,981

Math
if x = 1/y , then xy = 1 (x-1/x)(y+1/y) = (x^1 -1)/x * (y^2 + 1)/y = (x^1 - 1)(y^1 + 1)/xy = (1/y^2 - 1)(y^2 + 1)/1 = 1 + 1/y^2 - y^2 - 1 = 1/y^2 - y^2 or = ( 1 - y^4)/y^2 or more directly ( x- 1/x)(y+ 1/y) = xy + x/y - y/x - 1/(xy) = 1 + (1/y)/y - y/(1/y) - 1 = 1/y^2 - y^2 as...

Finate Mathematics
yes, easy to show for the y-intercept, x = 0 so in Ax + By + C = 0 0 + By = -C y = -C/B btw, what's with the (B,0) thing ?

math
Look at all those "Related Questions" identical to your below. I think the third one might be to your liking

math
consider the two chords PQ and PR the centre must be the intersection of the right-bisector of these two chords. midpoint of PQ= (-4,5) slope of PQ = 0 equation of right-bisector of PQ is x = -4 (no real work required for this one, notice that PQ was a horizontal line, so the ...

math
Part 1 : the centre of x^2 + y^2 = 25 is (0,0) the midpoint of the line from x to y is (0,0) , so the line runs through the centre as a diameter should Part 2: both of the given points satisfy the equation, so they must lie on the circle. So a line which falls on the circle an...

geometry
You have 6 identical equilateral triangles. the area of one of them = (1/2)(11)(11)sin60° = (121/2)(√3/2) so the are of the hexagon = 6(121/2)(√3/2) =appr 314 to the nearest whole number

math
If you end up with some x equals something, then there will be one solution e.g. #4 4(x+3)+2x = x-8 4x + 12 + 2x = x-8 5x = -20 ----> some x equals something x = -4 -----> one solution if your x term drops out, and you end up with a true statement, there will be an infin...

Math
multiples of 9: 9 18 27 36 45 54 63 72 81 90 .. multiples of 6 6 12 18 24 30 36 42 48 54 60 66 72 .. the ones found in each are: 18 36 54 72 .. that is multiples of 3*3*2 or 18 number of multiples of 18 in 1000: 1000/8 = 55.55.. There 55 of them

Math
Each of the 4 places can be filled with any of the 5 odd digits number of "odd-looking" number = 5*5*5*5 = 625 , assuming that digits my be repeated

Algebra
actually, any method should work for any of those conditions, in #3, I would use the intercept-intercept form if the x-intecept is 'a' and the y-intercept is 'b', then the equation can be written simply as x/a + y/b = 1 so x/6 + y/3 = 1 times 6 x + 2y = 6 , all...

Math
7.2 % per annum, compounded monthly corresponds to a rate of .072/12 or .006 per month so we have: 10000(1.006)^n = 25000 , where n is the number of months 1.006^n = 2.5 take log of both sides log (1.006)^n = log 2.5 n log 1.006 = log 2.5 n = log 2.5/log1.006 = appr 153 months

Algebra
Ahh, you mean directly? x/3 + y/6 = 1 (in general: x/(x-intercept) + y/(y-intercept) = 1 ) once you have that, multiply each term by 6, the LCD 2x + y = 6 check if both points lie on it. for (3,0) 2(3) + 0 = 6 , ok for (0,6) 0 + 6 = 6 , ok

Algebra
The x-intercept of 3 is really the point (3,0) and the y-intercept of 6 is really the point (0,6) So you are just given two points, find the slope, then find the equation of the line in your usual way.

3rd question - MATH
let Ø/2 = x for easier typing, then I should prove tanx = sin(2x)/( 1+ cos(2x)) RS = 2sinxcosx/(1 + 2cos^2 x - 1) = 2sinxcosx/(2cos^2 x) = sinx/cosx = tanx = LS

MATH
2. Again, I think you have a typo, and meant 2/(√3cosx + sinx) = sec(π/6 - x) RS = 1/cos(π/6-x) ) = 1/[ cos(π/6)cosx + sin(π/6)sinx ] = 1/[ (√3/2)cosx + (1/2)sinx ] = 1/( (1/2)(√3cosx + sinx) ) = 2/(√3cosx + sinx) = RS , if you had t...

MATH
First of all, to prove identities you have to work on either the left side (LS) or the right side(RS) independent of each other Secondly, .... 1. For your second line to be valid, the original should have been sec x + tan x( (1-sin^2 x)/cos x)=1 LS = 1/cos x + sin x/cos x(cos^...

math
t(n) = 3n - 4 , were n > 0, n ∈ I rate of change of t(n) is 3 rate of change of s(n) is 2 set t(n) = s(n) 3n-4 = 2 + 2n n = 6

Algebra
yes

Math
For the given data, the relation is y = 2x so, yes, the slope or the rate of change is 2

algebra
Since it is already factored for you, it now becomes trivial. By the "just look at it" theorem x = 2, happens once x = -3, happens twice x = 1/2 , happens four times, assuming the last part contains a typo, and you meant ..... (x-1/2)^4

trig
40° 5' = (40 + 5/60) = 481/12 ° You will have two right-angled triangle For the distance of the closer object tan (481/12)° = 57/d1 d1 = 57/tan(481/12)° For the distance of the farther away object tan 27 = 57/d2 d2 = 57/tan27° distance between objects =...

trig
After the 10 mile sailing, let the distance between boat and lighthouse be x by the cosine law, you can find x x^2 = 14^2+10^2 - 2(14)(10)cos57° , (90°-33°=57°) Once you have x, you can find a second angle using the sine law, and voila!, easy after that.

trig
I assume that the bearing is from the position of the ship. Then the angle opposite the 7 side is 18 degrees. Sin@/12 = sin18/7 Gives you the bottom angle

trig
I have helped you with 3 out of your 4 questions, they are all similar to each other. Give this one a try and let me know what you think.

trig
I am answering this from a Starbucks on my tablet and have no access to a calculator to check your answer.

trig
My diagram shows a triangle with sides 75 and 120 and a con-contained angle of 29° adjacent to the 75 side by sine law: sinØ/75 = sin29°/120 sinØ = 75sin29°/120 Once you have the second angle Ø , it is easy to find the third angle and answer yo...

math
rate = distance/time Alden: rate = 2/(3/4) mph = 2(4/3) = 8/3 mph or appr 2.67 mph Kira: rate = 1.75/.5 mph = 3.5 mph

Math
Is that supposed to be a vector? If so, then .... Squaring is basically the operation of multiplication. Since "multiplication" is not an operation that can be performed for vectors, the question makes no sense to me.

MATHS
look at the top post of "Related Questions" below

math
method 1: take 5% of 56.60 ... .05(56.6) = 2.83 Add that to the price of the shoes total cost = 56.60 + 2.83 = $59.43 Method 2 adding 5% to something is the same as taking 105% of the the 'something' and 105% = 1.05 so 1.05(56.060) = $59.43

MATH
let the sweets of Abigail and Ben be 4x and 5x ( note 4x : 5x = 4 : 5 ) A bit later .... Abigail has 4x + 16 Ben has 5x - 2 "Abigail has thrice as many sweets as Ben" ---> 4x+16 = 3(5x-2) I will let you finish it. (You should get Ben has 8 at the end, and Abigail ...

typ0 - math
B's rate ---- x L/min

math
A's rate --- 8 L/min B's rare --- x+8 L/min combined rate = (x+8) L/min for first 3 minutes, amount of water = 24 L next 5 minutes, amount of water = 5(x+8) or 5x+40 L so , 4 + 24 + 4x+40 = 77 4x = 9 x = 9/4 L/min or 2.25 L/min B's rate is 2.25 L/min

math
Let their respective amounts be m, a and b 30% of Muthu's share equals 80% of Ali's share .3m = .8a 3m = 8a m = 8a/3 Bill's share was 25% of Muthus share b = .25 m 4b = m Ali had $185 more than Bill a = b+185 or b = a-185 but since m = 8a/3 and m = 4b 8a/3 = 4b tim...

math
2(10^6) + 5(10^3) + 6(10^2) + 9(10) + 3 (not sure if your instructor would expect you to include the terms 0(10^5) + 0(10^4) , I wouldn't )

Algebra 1 (Reiny or Kuai)
elimination would be a poor choice for this one , why not use substitution ? y = (1/4)x into the first ..... 2x + 3(1/4)x = 33 times 4 8x + 3x = 132 11x = 132 x = 12 then y = (1/4)(12) = 3 x=12 , y = 3 by elimination: work on the 2nd 4y = x x - 4y = 0 times 2 2x - 8y = 0 so no...

Maths
To get the same answer is always a good thing. I also like the way our methods followed parallel tracks.

Maths
right now: Mary ---- x Wendy -- x+80 Jean --- x+30 after giveaway: Wendy --- (x+80)/2 Mary now has x + (x+80)/2 = (3x + 80)/2 but she gives (1/5)of that to Jean Mary is left with : (4/5)(3x+80)/2 and Jean now has (1/5)(3x+80)/2 + x+30 = (13x+380)/10 so now Jean has 62 more tha...

medical insurance
This is a strange question. Since Canada's universal health care covers all hospital services, like medication, beds, IV's etc, and covers all services performed by medical doctors, and covers most lab and diagnostic tests at 100% with no deductibles, the only one in y...

maths
I will assume you mean: 16 2/3 % of 16 = 50/3 % of 16 = (50/300)(16) = 8/3

WORLD GEOGRAPHY
I didn't know the answer, so I just googled "india's garden city" and got the same answer that Ms Sue just gave you.

trig
I assume you made a sketch. On mine I let the distance between tower and cliff be x m then tan60° = 100/x x = 100/tan60° In my smaller right-angled triangle at the top of my diagram , let the difference between height of tower and height of cliff be h then tan 30 = h/x...

Math
2. One of the state numbers must be exactly divisible by 6 and have an answer of 7 so (?) ÷ 6 = 7 which number fits that ? the rest are correct

Math
Neither, it would be -6 taking Steve's equation of y= x/2 + 3 , and letting y = 0 0 = x/2 + 3 -3 = x/2 -6 = x

Math
10 cm ---> 40 m 10cm ---> 4000 cm 1cm ----> 400 cm scale is 1 : 400

Math
Usually an inequality equation has two solutions ... |2x-2| = x 2x-2 = x or -2x + 2 = x , where x ≥ 0 x = 2 or -3x = -2 x = 2 or x = 2/3 both answers check out, so there are two solutions.

Math
for the x: (x+2)/(6+2) = 1/3 3x + 6 = 8 3x = 2 x = 2/3 for the y: (y-6)/(-8-6) = 1/3 3y - 18 = -14 3y = 4 y = 4/3 (x,y) = (2/3,4/3) check: distance form (-2,6) to (2/3,4/3) = √( (-2-2/3)^2 + (6-4/3)^2 = √( 64/9 + 196/3) =√(260/9) = (1/3)√260 distance fr...

Math
your given line of 2x + 8y = 10 has slope -2/8 or -1/4 so the slope of a perpendicular line is +4 its equation is y = 4x + b, but (5,0) lies on it, so 0 = 4(5) + b b = -20 new equation: y = 4x - 20 and the y-intercept is -20, (since we have it in slope y-intercept form)

Math
notice that in the place of x in f(x) you now find a -6 so in the expression , replace x with -6 as well f(x) = 7x^3+13x^3-3x^2 f(-6) = 7(-6)^3 + 13(-6)^3 - 3(-6)^2 = ... I suspect a typo, why would you have two different cubic terms ?

minor typy - Genius Math
I just notice two typos, but they don't affect the solution. in the bracketed angle , it should say (228-x) let angle ADC = x then angle BDC = 360-x-132 = 228-x in triangle ADC, by the sine law, DC/sin24 = AC/sinx DC = AC sin24/sinx in triangle BCD, DC/sin12 = BC/sin(228-x)

Genius Math
Since the original triangle is isosceles, it is easy to calculate that angle CAB =angle CBA = 42° then angle CAD = 24° and angle CBD = 12° Also angle ADB = 132° let angle ADC = x then angle BDC = 360-x-132 = 258-x in triangle ADC, by the sine law, DC/sin24 = AC...

math
see previous post Are you switching names ? why?

algebra
I will assume that by "simplify" you mean rationalize 5/(8+√7) = 5/(8+√7) * (8-√7)/(8-√7) = (40 - 5√7)/(64 - 7) = (40 - 5√7)/57

math
Since you can't show the diagram, I will surmise the following inner radius --- x outer radius --- x+1 area of onion ring = π(x+1)^2 - x^2) = π(x^2 + 2x + 1 - x^2) = π(2x+1) , looks like the first choice

math
done, see your other post

cal
What methods have you been taught for this kind of integration ? Do you know how to do integration by parts ?

math
just add up their circumferences. recall that circumference of a circle = π(diameter) so we have 3π + 3.2π + 2.8π + 3.5π + 2.9π = π(3+3.2+2.8+3.5+2.9) = .....

cal
You can always check an integral by differentiating your result. If I do that to your answer I get: 25(secx + tanx)^4 ( secxtanx + sec^2 x) I doubt very much if that can be reduced to a simple 5(secx)^4 Wolfram answer can be reduced to : (5/3)tanx (2 + sec^2 x) http://integral...

cal
oops, sorry about dropping that π in the 4th last line but, I don't see why it should not be π(e^2 + 1) check my arithmetic.

cal
One of the integrals you should have in your repertoire of common integrals is ∫lnx = xlnx - x so volume = π∫ lnx dx from x-1 to e^2 = xln - x | from 1 to e^2 = e^2(lne^2) - e^2 - (1ln1 - 1) = e^2(2) - e^2 - 0 + 1 = e^2 + 1

math
Why don't you just type it using the language of arithmetic? did you mean: (4/5) ÷ (8/3) -1/5 ? then = (4/5)(3/8) - 1/5 = 3/10 - 2/10 = 1/10 remember , when dividing one fraction by another write down the first fraction, TIMES the reciprocal of the second. That is w...

math
What step by step ? Just multiply by 2 -6 < 2x < 0

Algebra 1B
I will assume that you have a typo and you meant - 1.6t^2 - 28t + 200 since it would only decrease for the first 8 or 9 minutes, after that the heart rate would actually increase. This makes no sense. which is it ?

geometry
Did you sketch it ? AB= √(0^2 + 8^2) = 8 F is (5/2,8) E is (5/2, 0) FE = √( (5/2-5/2)^2 + (8-0)^2 ) = √(0 + 64) = 8 so AB = EF

math
Yup, found it. Rats! Hope Frank finds it.

math
3a+3(b+1)=c 3(b+1) = c - 3a b+1 = (c-3a)/3 or c/3 - 3

Math grade 9
not sure about grade 9, but I just checked an old grade 10 text from Ontario, and I used to teach it in 10 The proof followed your method using a general case.

Math grade 9
If (a,b) is a point not on the line Ax + By + C = 0 then the shortest distance from the point to the line is |aA + bB + C|/√(A^2+b^2) shortest distance = |5(1) + 6(1) -3|/√(1^2+1^2) = 8/√2 which rationalizes to 4√2

Math
Wouln't it automatically be a right-angled triangle if the x-axis and the y-axis form sides? So let's find the equation of the hypptenuse. m = -3, point is (1,3) y = -3x + b 3 = -3(1) + b b = 6 y = -3x + 6 base of triangle = x-intercept = 2 height of triangle = y-inter...

Math
did you sketch it? did you get something like http://www.wolframalpha.com/input/?i=plot+y+%3D+%7Cx%7C%2B1+%2C+y+%3D+5 notice that they intersect at (4,5) and (-4,5) ? and the altitude is 4, and the base is 8 Area = (1/2)(8)(4) = 16

Advanced Functions/ Precalculus Log
remember if the base of the log is 10, we don't have to write the base. so you have y= logx and y = 2log(x-4)+3 intersection: 2log(x-4) + 3 = logx log(x-4)^2 - logx = -3 log ( (x-4)^2 /x ) = -3 (x-4)^2 /x = 10^-3 x^2 - 8x + 16 = (1/1000)x 1000x^2 - 8001x + 16000=0 using t...

Math. Urgent please help
draw the altitude from A to form a right-angled triangle. Cos C = 4/16 = 1/4 In triangle BCD, by the cosine law ... BD^2 =8^2 + 8^2 - 2(8)(8)cosC = 64 +64 - 128(1/4) = 160 BD = √160 = 4√10 Where does the centroid G enter the picture?

Last two CALC questions. :)
8. f'(x) = -5cos(7-5x) f'(π) = -5cos(7-5π) = 3.769 or your last choice of 3.770

Calc help - check rest of answers, please?
I did #4 and #5 for you in your previous post #6 let the width of the base be x let the length of the base be 2x let the height be y given: Volume = 10 2x^2y = 10 y = 5/x^2 which was the last choice. cost = 10(2x^2) + 6( 2 ends + front + back) = 20x^2 + 6(2xy + 2xy + 2xy) = 20...

CALCULUS - Check my answers :)
#5 At a time of t seconds after the boy passed under the balloon, distance covered by boy = 15t height of balloon = 5 + 5t let the distance between them be d ft d^2 = (5t+5)^2 + (15t)^2 2d dd/dt = 2(5t+5)(5) + 2(15t)(15) dd/dt =( 5(5t+5) + 15(15t) )/d when t = 3, d^2 = 20^2 + ...

CALCULUS - Check my answers :)
#4, the way you typed it, ... y' = 2x/3 when x = 8, y' = 16/3, which is not found in any of your choices as the slope if you meant , f(x) = x^(2/3) then y' = (2/3)x^(-1/3) = (2/3)(1/x^(1/3)) when x = 8 y' = (2/3)(1/2) = 1/3 the only one that has a slope of 1/2 ...

math
yes, but be more confident in your choice

math
Treat these as if they were equations but keeping the in-equality sign instead of the equal sign, EXCEPT when you multiply or divide the in-equation by a negative, you must reverse the inequality sign. e.g. last one: -8x > -64 I would have to divide by -8 to get +x , so x &...

math
No, you did not follow the instructions, it says ... take the square root of each number , THEN add them up, so √a + √36 + √ 16 + √9 = 15 √a + 6 + 4 + 3 = 15 √a = 2 a = 4

math
How about you giving this one a try ? Just follow the steps they told you to take, let me know what you get, and how you got it.

math
Then 8^z + 10^z + 11^z = 3 by the "just look at it" theorem, z = 0 since (anything)^0 = 1

math
so just follow the instructions, ((1)(2)(3)z)^2 = 144 (6z)^2 = 144 36z^2 = 144 z^2 = 4 z = 2 Actually the wording of the question is flawed. Terms are separated by addition and subraction, there is only one term here. The correct wording should be : If #(w,x,y,z)means you squa...

math
If the scale factor is 3, then A(2,0) ----> A' (6,0) B(3,-1) ---> B' (9,-3) C(-2,-5) --> C' (-6,-15) Can't get any simpler than that. Might be a good idea to graph the original triangle and the new triangle. What do you notice ?

geometry
I sketched your triangle and concluded that the right angle must be at N It is easy to show that triangle MPN ≈ triangle NPO then MP/PN = PN/PO 15/9 = 9/PO 15PO = 81 PO = 81/15 = 27/5

Coordinates in 3-D, Polar Coordinates, Precalculus
Have some fun here, changing the values of a and b I used a=4 and b=3 http://www.wolframalpha.com/input/?i=plot+r+%3D4cos%28Ø%2B3%29+%2C-6.28+%3C+Ø+%3C+6.28 Notice I also placed a domain on Ø from -2π to +2π things really get interesting if you p...

math
number of correct answers --- x number of incorrect answers --- 150-x 2x - .5(150-x) = 270 2x - 75 + .5x = 270 2.5x = 345 x = 138 He/she should answer at least 138 questions correctly to have score of at least 270 check: 138(2) - 12(.5) = 270

math
let the payment be P P (1.035^16 - 1)/.035 = 100000 solve for P on your calculator you should get around ($4770 , not the exact answer so you will still have to do the calculation yourself)

Vectors
The plane must pass through the midpoint of the given points which is (3,10,6) A direction vector through the 2 points is [2, 4, 6] or reduced to [1,2,3] This must be a normal to the plane, thus the equation of the plane would be x + 2y + 3z = c but (3,10,6) must lie on it, so...

college algebra
multiply 2nd by 8 8x - 288y = -2640 subtract that from the first 297y = 2673 y = 9 sub into 1st 8x + 81 = 33 8x = -48 x = -6

Math
let the first odd whole number be n-8 then the others are n-6, n-4 , n-2 , n , n+2, n+4, n+6, and n+8 their sum = 9n (now wasn't that a clever way to define my numbers?) 9n = 243 n = 27 first = 27-8 last = 27 + 8 and that sum = 27+27 = 54

Math
I will assume a typo, and "treaters" is "greater" let the unit digit of the number be x let the tens digit of the number by y so the number is 10y+x and the product of the digits is xy when is xy > 10y+x xy - 10y > x y(x-10) > x y > x/(x-10) but...

math
Are you counting 1 as a factor ? 12 has a lot of factors, let's look at it 12 has factors 1,2,3,4, 6, and 12 , six of them none other comes close. I would say 12 if we exclude 1 as a factor.

Math
then log2 ((3x-1)/x) = -2 2^-2 = (3x-1)/x 1/4 = (3x-1)/x 12x - 4 = x 11x = 4 x = 4/11

MATH-MS.SUE PLEASE HELP
Maria sells 3 cookies per day Stephen sells 6 cookies per day Aisha sells7 cookies per day together they sell 16 cookies per day number of cookies needed to make $320 = 320/2 = 160 number of days needed to sell 160 cookies at a rate of 16 cookies per day = 160/16 = 10 days

Math
Did you observe how the numbers change ? +1 , -2 , +4 , -8 , +16 , ...... Can you see the pattern? Can you state the next number in your sequence ? let me know what you get.

math
I think we must assume the race track is a circle. Let's work in feet, there are 5280 feet in a mile, since it takes 3 laps to make a mile each lap is 5280/3 or 1760 ft for one lap. So the circumference of the race track is 1760 feet.

AP CALC
Did you notice that your given point does not lie on your curve ? Is there a typo? Usually for this type of question, the given point lies on the given curve. If that is what you intended , please confirm this before I attempt what will be a very nasty question

Math
Let the points scored by Alan, Gil and Deb be 2x, 3x, and 4x , .......... (notice that is in the ratio of 2:3:4 ) so 4x = 114400 x = 28600 then 2x = 57200 and 3x = 85800 Alan scored 57200 Gil scored 85800 Deb scored 114400 Add them up to get the total score.

Math
1st one, in your head, a = 4 or a = -2 2. |2a+5| > 3 2a+5 > 3 OR -2a -5> 3 a > -2 OR a < -4 3. looks like a typo , I see neither an equation nor an in-equation. If it is one of those, first divide both sides by 3, then it is very easy.

Math
Nina, why do you keep reposting this question. I see from the Related Questions below, that it was successfully answered for you by Ms Sue.

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