Sunday

February 7, 2016
Total # Posts: 32,824

**algebra**

So you want the base of a right-angled triangle, call it x x^2 + 80^2 = 100^2 continue ...
*October 26, 2015*

**math**

So the inner dimensions, which determine the volume are: 8-1 or 7 cm 4 cm and 3 cm volume = 7x4x3 cm^3 = 84 cm^3
*October 26, 2015*

**maths**

Write down the 7 days of the week. Count the number of days that begin with the letter T. all yours ....
*October 26, 2015*

**maths**

Are we talking about the 7 days of the week, with Friday being one of them ? I think I just gave you the answer.
*October 26, 2015*

**Math word problem**

you want the absolute value of the difference between the two numbers, that is ... |-11.6 - (-5.43)| =| -6.17 | = 6.17 to understand, place -11.6 and -5.43 on a number line. How far is the distance between you two markings ?
*October 26, 2015*

**Maths**

Unless you specify some kind of border, you just have the perimeter of a rectangle with length 30 cm and width 20 cm. Just add up the 4 sides. Make a sketch to help you to visualize the problem.
*October 26, 2015*

**vectors linear algebra**

It is easy to find the equation of the L2, since it is parallel to L1 and you have a point on it direction vector of L1 is (-6,2,-4) or reduced: (3, -1, 2) a possible line equation for L2 is x = -3 + 3k y = -3 - k z = -3 + 2k you want x = -12, so -12 = -3 + 3k 3k = -9 k = -3 ...
*October 26, 2015*

**Math**

John's age --- x David's present age --- x+3 in 8 yrs: John = x+8 David = x+3 + 8 = x+11 x+8 = (7/8)(x+11) solve for x and finish
*October 25, 2015*

**Math**

let's first of all assume that the two 1's are distinguishable, e.g. one is red, the other blue case1, our even number ends in a zero number of such cases = 6x5x4x3x2x1x= 6! = 720 but the two 1's ARE distinguishable, so the number of evens that end in a zero are ...
*October 25, 2015*

**Math- word prolems**

let radius be r inches let the height be h inches vol = πr^2h 166.25 = πr^2 h h = 166.25/(πr^2) "economical cylinder" is a function of the surface area, SA SA = 2 circles + rectangles of the side cost = 2(2πr^2) + 6(2πrh) = 4πr^2 + 12&#...
*October 25, 2015*

**math**

geeeesh! adding up 5 numbers ??
*October 25, 2015*

**probability**

You don't state what the prob of heads or tails are of the biased coin. Just to show the method of my calculations, I will assume the prob(heads) = 3/5 and prob(tails) = 2/5 Make yourself an addition and a multiplication table for 1,2,3,4 In the 16 possible sums, many are ...
*October 25, 2015*

**Math Answer Check**

csc^2 x = 5 or sin^2 x = 1/5 sin x = ± 1/√5 So x is in all 4 quadrants. the reference angle is 26.57° so x = 26.57° x = 180-26.57 = 153.43° x = 180 + 26.57 = 206.57° or x = 360-26.57 = 333.43° just notice that your domain is 0 ≤ x ≤...
*October 24, 2015*

**algebra**

let the number be x x-3 = 39-x 2x = 42 x = 21
*October 24, 2015*

**Algebra**

I read that as |R-136| ≤ 20 -20 ≤ R-136 ≤ 20 add 136 116 ≤ R ≤ 156 you are correct
*October 24, 2015*

**math**

Isaac's law says, T(t) = Ta + (Ts - Ta)e^kt where Ta is the room temp Ts is the temp of the object and t is the time T(6) = 72 + (173-72)e^(6k) 137 = 72 + 101e^6k 101e^6k = 65 e^6k = 65/101 6k = ln(65/101) k = ln(65/101)/6 = -.073455541.. (put in calculator's memory) ...
*October 24, 2015*

**Mathematics**

in a GP term(2) = ar = -5 term(5) = ar^3 = 48 ar^3/(ar) = -48/5 r^2 = -9.6 no real solution possible
*October 24, 2015*

**Mathematics**

in a GP a = 5 term(3) = ar^2 = 5r^2 = 80 r^2 = 16 r = ± 4 if r = +4 , term(4) = ar^3 = 5(4^3) = 320 if r = -4 , term(4) = ar^3 = 4(-4)^3 = -320 check: if a = 5, r = 4 first 5 terms are: 5, 20, 80, 320, 1280 if a = 5, r = -4 first 5 terms are: 5, -20, 80, -320, 1280 all ...
*October 24, 2015*

**Mathematics**

given: a = (1/2)d or d/2 term(8) = a + 7d = d/2 + 7d = 15d/2
*October 24, 2015*

**Calculus help plz**

The first part is somewhat vague. Are you finding the derivative of 8(x-y)^12 with respect to x , with respect to y, with respect to ? assuming "with respect to x", d(8(x-y)^12)/dx = 96(x-y)^11 (1 - dy/dx) your second question: y^x + x^y = 17 and x+y=5 Since this ...
*October 24, 2015*

**math**

I did the same question about a week ago, all you have to do is change a few numbers. Without doing the actual steps again, watch if near the end, LS = RS , then they do intersect, and you can find the point or LS ≠ RS, then they would NOT intersect
*October 24, 2015*

**simple interest rate**

I = PRT 200 = 5000(R)(1) R = 200/5000 = .04 or 4%
*October 24, 2015*

**algebra**

.036(57^2) - 2.8(57) = 62.56 = 116.964 - 159.6 + 62.56 = 19.924 , but that is in thousands, so .... 19924 looks like you couldn't read your own writing, look at your first term
*October 24, 2015*

**algebra**

Does it sound plausible to you that in an entire country only 65 died in one year ? how did you possible get 65? show me your steps so I can see where you went wrong.
*October 24, 2015*

**algebra**

just plug x = 57 into your equation and evaluate. Surely you have a calculator. you probably meant: y=0.036x^2 −2.8x+62.56 , remember to square 57 first, then multiply by .036 Also be mindful that y is per thousand people, so you have to multiply your final answer by ...
*October 24, 2015*

**math**

using sum(n) = (n/2)(2a + (n-1)d) and a = 1 sum(20) = 10(2 + 19d) = 20 + 190d sum(40) = 20(2 + 39d) = 40 + 780d sum of next 20 terms = sum(40) - sum(20) = 40+780d - 20 - 190d = 20 + 590d 20+190d = (1/3)(20+590d) 60 + 570d = 20+590d 40 = 20d d= 2 sum(30) = 15(2 + 29d) = 30 + ...
*October 24, 2015*

**Calculus 1**

Already factored, nice So all we have to do is set each factor equal to zero x^(4/5) = 0 ----> x = 0 , f(0) = 0 (x-1)^2 = 0 ----> x = 1 , f(1) = 0 F ' (x) = x^(4/5) (2(x-1)) + (4/5)x^(-1/5) (x-1)^2 = (2/5)x^(-1/5) (x-1) [5x + 2(x-1)] = (2/5)(x-1)(7x-2)/x^(1/5 = 0 for...
*October 23, 2015*

**maths**

unit digit --- x ten's digit ---- 4x clearly obvious that x can only be 1 or 2 So the number could be only 41 or 82 and the number with the digits reversed of those is 14 and 28 which has a difference of 54 ?
*October 23, 2015*

**maths**

daughter's share = 1 - 1/2 - 1/3 = 1/6 (1/6)x = 5000 x = 30000 take over
*October 23, 2015*

**Algebra**

Now you have changed the question from your previous post. If all you know is the perimeter to be 172, the answer would not be unique e.g. suppose the rectangle is 50 and 36 ft (notice perimeter is 172) area = 50(36) or 1800 ft^2, thus we need 1800 tiles e.g. suppose the ...
*October 23, 2015*

**Algebra**

No, you are filling the rectangle with the tiles, so we are talking about AREA. area = 60 x 26 = 1560 so you need 1560 tiles (the fact that the perimeter is 172 ft was not needed, we could have calculated that anyway)
*October 23, 2015*

**algebra**

you are just finding the hypotenuse of a right-angled triangle with sides l and 7l Hyp^2 = l^2 + (7l)^2 = 50l^2 hyp = √(50l^2) = 5√2 l
*October 23, 2015*

**MATH**

solve for x (1/2)x - 9 = 14 you should be able to solve this equation in your head.
*October 22, 2015*

**Math, not Prabhat academy**

Did you make a sketch? If so, it should be intuitively obvious that the angle must be 60°, since the resultant will form an equilateral triangle with the two given vectors
*October 22, 2015*

**Math**

rate of reading at the beginning = 40 pages/hr at that rate the time to read 150 pages = 150/40 hrs or 15/4 hours Time left to read the other 250 pages = 6-15/4 or 9/4 hrs rate of reading needed to finish the book = 250/(9/4) = 111 1/9 pages per hour or appr 1.85 pages per minute
*October 22, 2015*

**Math**

f(x) = 4√(√x + 6) = 4(2 x^(1/2) + 6)^(1/2) f ' (x) = 2(2x^(1/2) + 6)^(-1/2) (x^(-1/2) ) = 2/(√x(2√x + 6) )
*October 22, 2015*

**algebra**

for a reflection over the x-axis, the x's stay the same, but the y's change to their opposites. I will do point B B(3,-2) ----> B' (3,2) you do the rest. I suggest to make a sketch to confirm what you did
*October 22, 2015*

**Math**

yes
*October 22, 2015*

**algebra**

Tricky question . Are we dealing with a parabola? I will assume we are. Since the axis of symmetry is neither horizontal nor vertical, we got a mess. Before I start, I want to make sure you did not have a typo. Let me know.
*October 22, 2015*

**Albegra**

number of adult tickets ---- x number of children's tickets ---- 325-x solve for x ..... 3.5x + 2(325-x) = 875
*October 22, 2015*

**Algebra**

number of dimes --- x number of quarters = x+3 number of nickels = x-5 solve for x x + x+3 + x-5 = 52
*October 22, 2015*

**Math**

The best webpage for this kind of problem on the entire internet Just enter you data http://davidmlane.com/hyperstat/z_table.html
*October 22, 2015*

**MATH**

amount given by Maya to Emily --- x after the give-away: Maya ---- 2740 - x Emily ---- 3560 + x 3560 + x = 4(2740 - x) take over
*October 22, 2015*

**math**

Let's hope they are on separate tracks. let distance covered by faster train be x miles then distance covered by slower train is 800-x When they pass each other, their times must be the same, so ... x/67 = (800-x)/53 53x = 53600 - 67x 120x = 53600 x = 446 2/3 miles time = ...
*October 22, 2015*

**mathematical statistic**

2 initial cases: case 1, the ball drawn from urn A to B is W urn B now has 8W and 5B prob (W and W) = prob(W from A AND W from B) = (6/9)(8/13) = 48/117 rest of problem to check my answer: prob(W and B) = (6/9)(5/13) = 30/117 case2: the ball from urn A to urn B is B Prob(B and...
*October 22, 2015*

**Calculus**

let the width of the photo be x in then the length of the photo is 128/x in width of poster = x+8 length of poster = 128/x + 8 Area = (x+8)(128/x + 8) = 128 + 8x + 1024/x + 64 d(area) = 8 - 1024/x^2 = 0 for a min of area 8 = 1024/x 8x^2 = 1024 x^2 =128 x = √128 = 8√...
*October 21, 2015*

**Math (Trig)**

distance in the direction of the bearing = 6(15) or 90 n miles. make your sketch to see: sin 25° = x/90 x = 90sin25 = appr 38.04 n miles cos25= y/90 y = 90cos25= 81.57 n miles
*October 21, 2015*

**Math**

according to your explanation, #1 is 120 ≤ (135+145+n)/3 ≤ 145 120 ≤ (280 + n)/3 ≤ 145 times 3 360 ≤ 280 + n ≤ 435 subtract 280 80 ≤ n ≤ 155 do the others the same way
*October 21, 2015*

**Math**

so we have (-1 + i√3)^12 let u = -1 + i√3 = √10(cos(2π/3) + i sin(2π/3)) u^12 = √10^12(cos 8π + i sin 8π) = 10^6( 1 + i(0)) = 10^6 or 1,000,000
*October 21, 2015*

**Math**

1 D is correct For #2, you answer of B wouldn't do anything to the point, that is what you started with reflections in the x-axis leave the x alone but change the y to the opposite, so (6,-2) ---< (6,2)
*October 21, 2015*

**math**

work the problem in one set of units, e.g. cm All your really need to know : 1 inch = 2.54 cm Annalisa = 64 inches = 162.56 cm Stefan is 7.5 cm taller , so Stefan = 170.06 cm Keiko is 1.5 inches or 3.81 cm shorter than Stefan Keiko = 166.25 To convert to inches, just divide ...
*October 19, 2015*

**math**

umhhh, why not look back at your earlier posts before reposting the same question ? http://www.jiskha.com/display.cgi?id=1445204064
*October 18, 2015*

**math**

for L1, direction vector is (3,-1,-1) and an equation is x = 1 + 3t y = -t z = -1 - t for L2, a possible equation is x = -6 - 3k y = 21 + 9k z = -8 - 3k if they intersect, 1+3t = -6-3k AND -t = 21+9k AND -1-t = -8-3k using the first two equations, 1 + 3(-21-9k) = -6-3k -62 - ...
*October 18, 2015*

**Complete the Square**

The hardest type have fractional coefficients that are relatively prime. I will give you one, and solve it, you pick 4 more like it y = (2/3)x^2 + (5/7)x - 11/13 factor the coefficient (divide it out) of the x^2 terms from the first two terms, leave the constant term trailing ...
*October 18, 2015*

**algebra**

I tried factors of 6 and found f(-2)=0 and f(3) =0 so after successive synthetic divisions I had the factors, (x+2)(x-3)(7x^2 - 2x - 1) solving (x+2)(x-3)(7x^2 - 2x - 1) = 0 x = -2, 3, or (2 ±√32)/14 = -2, 3, or (1 ± 2√2)/7
*October 17, 2015*

**Math (help)**

f(1) = 0 , after the first try so x-1 is one of the factors, by synthetic division we have (x-1)(2x^3 + x^2 + 8x + 4) , notice we can group the last one =(x-1)( x^2(2x+1) + 4(2x+1) ) = (x-1)(2x+1)(x^2 + 4) = 0 for zeros of the function, so x = 1 , -1/2 , or ±2i
*October 17, 2015*

**St Eugene university**

First of all: St Eugene university is not a school subject second: I can't make any sense out of your post. Is it: x^2 * (second derivative of -3x) * (first derivative of -5y) = sin(logx) ?? is it the derivative with respect to x ? then 2nd derivative of -3x is 0, and you ...
*October 17, 2015*

**Luhlaza high school**

a = 2 v = 2t + c , where c is a constant and t is in seconds when t = 0 , v = 0 0 = 0(t) + c ----> c = 0 so v = 2t after 10 s, v = 2(10) or 20 m/s
*October 17, 2015*

**Math**

William's age --- x Jon's age ------ x+6 in 2002: x+6 = 4(x-5) + 2 x+6 = 4x - 20 + 2 -3x = -24 x = 8 so william is 8 in 2002 and Jon is 14 in 2002 you do the rest
*October 17, 2015*

**Probability**

prob(rotten) = 2/2 = 1/6 prob(not rotten) = 5/6 You want prob that none of 4 are rotten = C(4,0)(1/6)^0 (5/6)^4 = 1(1)(625/1296) = 625/1296
*October 16, 2015*

**math**

Quite easy .... Make a sketch, draw in the altitude, and mark the centre. The radius is 12 cm, and the distance from the centre to the triangle must then be 6 cm You have a right-angled triangle with one leg 6, the hypotenuse as 12 and a missing base, call it x x^2 + 6^2 =12^2...
*October 16, 2015*

**Calculus**

let each side of the square that you cut out be x " Recall that volume of a box = length x width x height Make a sketch showing the cut-outs, you will see that length = 32-2w width = 24-2w height = w V = w(32-2w)(24-2w) = 768w - 112w^2 + 4w^3 dV/dw = 768 - 224w + 12w^2 = ...
*October 16, 2015*

**math**

amount at end of 3 years = 4000+960 = ... I = PRT 960 = 4000(R)(3) R = 960/12000 = .08 or 8% had the interest been 9% I = 4000(.09)(3) = 1080 take over
*October 16, 2015*

**Maths**

I am sure you meant 6% each year, not each day (at per day, the answer would be an absurdly large number) after 1 year -- 20000(1.06) after 2 years -- 20000(1.06)^2 .. after 8 years: amount = 20000(1.06)^8 = ....
*October 16, 2015*

**math**

Make a sketch Clearly the ship with the smaller angle is farther away. Label the two ships A and B, B closer to the lighthouse. Label the top and bottom of the lighthouse P and Q. enter the angles and AB=200 in triangle ABP, I have angle A = 30° angle ABP = 135° angle ...
*October 16, 2015*

**business maths**

count the days Interest = 720(.07)(days/365) =
*October 16, 2015*

**maths**

proof of what Justin said: let sides of original square be x by x perimeter = 4x new length = x+3 new width = x-3 new perimeter = 2(x+3) + 2(x-3) = 2x + 6 + 2x - 6 = 4x , just as before so original side = 84/4 = 21
*October 16, 2015*

**maths**

now: width --- x length -- x+7 after change: width = x-3 length = x+7 - 10 = x-3 (notice the new rectangle is now a square) original area = x(x+7) new area = (x-3)(x-3) the difference is 95 , so .... x(x+7) - (x-3)^2 = 95 x^2 + 7x - (x^2 - 6x + 9) = 95 carry on, it is easy
*October 16, 2015*

**Algebra**

amount at 6% ---- x amount at 7% --- 29000-x solve for x .06x + .07(29000-x) = 1895
*October 16, 2015*

**math**

4sinØ = -1, 0° ≤ Ø ≤ 360° sinØ = -1/4 my calculator gives me Ø = -14.48° but that is outside our domain Here is how I do those types: Since sinØ is negative, I know that my angle Ø must be in quadrants III or IV...
*October 16, 2015*

**Algebra**

Plug it in y = -2x -4 = -2(1) -4 = -2 is that a true statement? If it is, then your point is a solution, otherwise ....
*October 16, 2015*

**math**

I should have pointed out that there are two types of deviation. One is called Population Standard Deviation, where you divide by n, the number of data values The other is the Standard Deviation, where you divide by n-1 , where n is the number of data values. You can see ...
*October 16, 2015*

**math**

https://www.easycalculation.com/statistics/standard-deviation.php
*October 16, 2015*

**Algebra**

#4 but you typed 2/8, not 2/9 that is why both me and Bob labeled it incorrect. yes, the correct answer is 2/9 #3 so we have 3/4 wall in 1 hour or 3/4 wall in 60 minutes thus (3/4)÷60 or 1/80 of a wall in 1 minute
*October 16, 2015*

**Algebra**

#3 probably contains a typo, and it must have asked for the rate per hour, not per minute She did paint 3/4 of the wall in 1 HOUR. #4, check your calculation, you are close. You did multiply your quotient answer by 2 ??
*October 16, 2015*

**Trig**

length --- x width = x+9 height = x-1 volume = x(x-1)(x+9) This is not trig.
*October 15, 2015*

**Math(HELP!)**

let the length parallel to the barn be y ft let the two other sides be x ft each so 2x + y = 120 y = 120 - 2x area = xy = x(120-2x) = -2x^2 + 120x this is a downward parabola, the x of the vertex is -b/2a = -120/-4 = 30 then y = 120-60 = 60 So the largest area is 30(60) or ...
*October 15, 2015*

**algebra**

and ??
*October 15, 2015*

**Pre-Calculus**

An hour?? Did you make a sketch? Mine is a right-angled triangle with base angle of 40°, a horizontal of x, and a vertical of 75 ft, and a hypotenuse of h "How far out is the ship from the lighthouse" is a vague question. Do you want the direct distance between ...
*October 15, 2015*

**Math**

in the simplification process you would have to state it this way .... y=(x^2+x-12)/(x+4) = (x-3)(x+4)/(x+4) = x - 3 , x ≠ -4 So in effect y=(x^2+x-12)/(x+4) is the same as y = x - 3 for every value of x except -4 If we sub x = -4 into y = x-3, we get y = -7 If we sub x...
*October 15, 2015*

**algebra 1**

Did you translate the English into Math ? "nine tenths of a number g" ---> (9/10)g "is the same as" ----> = "seven plus two thirds of another number k " ---> 7 + (2/3)k follow the same steps I showed you many posts ago let me know what ...
*October 15, 2015*

**algebra 1**

not possible t - 7 = t -7 = 0 ----> contradiction
*October 15, 2015*

**HELPPP**

Really ?? http://www.jiskha.com/display.cgi?id=1444950413 http://www.jiskha.com/display.cgi?id=1444950445 http://www.jiskha.com/display.cgi?id=1444951131
*October 15, 2015*

**algebra 1**

follow my method I showed you in the previous ones I did for you. Show me your work
*October 15, 2015*

**algebra 1**

Why are you having difficulties with these? They all seem like the same type. (df + 10)/6 = g times 6 df + 10 = 6g subtracct 6 df = 6g - 10 divide both sides by d f = (6g-10)/d OR 6g/d - 10/d
*October 15, 2015*

**algebra 1**

10m = p - N m = (p - N)/10
*October 15, 2015*

**Pre-Calc**

Just translate the English into Math Ming's age --- x Jeffs age is 3x Tina's age = 2x+4 next year: Jeff's age = 3x+1 Ming's age = x+1 It said: 3x+1 = 2(x+1) + 7 3x+1 = 2x+2+7 x = 10 So today: Ming is 10 Jeff is 30 Tina is 24 A. So it looks like today, Tina is 2...
*October 15, 2015*

**Pre calc- confused**

present area = 250(160) = 40000 new area = 1.5(40000) = 60000 let the new length be 250+x let the new width be 160+x (250+x)(160+x) = 60000 solve for x, Hint, use the quadratic equation, since it does not factor.
*October 15, 2015*

**Math(HELP!)**

Did you get #1 ?
*October 15, 2015*

**Integral so difficult i need help**

Even Wolfram had trouble with it http://www.wolframalpha.com/input/?i=integrate+x%5Ex
*October 15, 2015*

**Math**

let prob(tail) = x let prob(heads) = 2x , (it said so in the question) but x + 2x = 1 x = 1/3 prob(tails) = x = 1/3 prob(heads) = 2x = 2/3 prob(exactly 2 out of 3 tails) = C(3,2) (1/3)^2 (2/3) = 2/9
*October 15, 2015*

**Math**

subtraction does not enter the picture So you want: Prob(floor polisher ok AND the washing machine not ok) prob(washing machine not ok) = 1 - .82 = .18 so Prob(floor polisher ok AND the washing machine not ok) = (.78)(.18) = .1404
*October 15, 2015*

**geomtry**

solve x + x+24 = 180
*October 15, 2015*

**Math**

number of games = C(5,2) = 10 just confirming the answer with concept of combinations.
*October 15, 2015*

**math**

number of coats --- n purchase cost per coat = 720/n total return = 40n profit = 40n - 720 cost of 8 coats = 8(720/n) 40n - 720 = 8(720/n) 40n^2 - 720n - 5760 = 0 n2 - 18n - 144 = 0 (n - 24)(n + 6) = 0 n = 24 or n=-6, rejecting the negative He bought 24 coats check: cost per ...
*October 15, 2015*

**MATHS**

W+6TW=11-W 2W + 6TW = 11 W(2 + 6T) = 11 W = 11/(2 + 6T) r=9-7jk+4jt^2 4jt^2 - 7jk = r - 9 j(4t^2 - 7k) = r - 9 j = (r - 9)/(4t^2 - 7k)
*October 15, 2015*

**Calculus**

Your first-line derivative is correct. g'(x) = (x^2+1)^5[6(x^2+2)^5(2x)] + (x^2+2)^6[5(x^2+1)^4(2x)] = 12x(x^2+1)^5 (x^2+2)^5 + 10x(x^2+1)^4 (x^2+2)^6 = 2x(x^2+1)^4 (x^2+2)^5 [6(x^2+1) + 5(x^2+2)] = 2x(x^2+1)^4 (x^2+2)^5 (6x^2 + 6 + 5x^2 + 10) = 2x(x^2+1)^4 (x^2+2)^5 (11x^...
*October 14, 2015*

**math**

George had the same question yesterday http://www.jiskha.com/display.cgi?id=1444046531
*October 6, 2015*

**Statistics**

P(A and B) = P(A) + P(B) - P(A or B) =.4 + .2 - .5 = .1 P(A|B) is a conditional probability Of the top of my head I am not really sure of the formula. I would have to Google it, but it is easy to find 2. Mutually exclusive means that the result of one event has not effect on ...
*October 6, 2015*

**Calculus**

you v(t) and a(t) are both correct sin 2π/3 = √3/2 cos 2π/3 = 1/2 so 8(√3/2) = 4√3 = appr 6.92820323 on my calculator so it depends what accuracy is required both of your answers are correct to two decimal places.
*October 6, 2015*