# Posts by STEVE

Total # Posts: 51,221

**Trigonometry**

the domain must be at most 1/2 period, or 2?/.1 = 20?. In addition, it must not contain an asymptote or a max/min, because there the curve doubles back, so it fails the horizontal-line test. All of those intervals are short enough, so we need to consider that there is an ...

**math**

draw a diagram of the corner of the field. Since the rope is shorter than either dimension, the grazed area is just a 1/4 circle. You know how to figure the area of a circle, right?

**math**

cot sec^2 - cot = cot(sec^2-1) = cot(-tan^2) = -tan sin tan + cos = sin*sin/cos + cos = (sin^2 + cos^2)/cos = 1/cos = sec so, doing the division, you have -tan/sec = -sin/cos * cos = -sin = -cos tan^3 ...

**Calculus**

as with the other problem, y = ?[?x,x^3] 10?t sin(t)dt y' = (10?(x^3) sin(x^3))(3x^2) - (10?(?x) sin(?x))(1/(2?x)) = 30x7/2 sin(x^3) - 5/?x sin(?x)

**Caclulus**

the 2nd FT of Calculus is just the chain rule in reverse. If F(x) = ?f(x) dx ?[u,v] f(t) dt = F(u)-F(v) so, taking derivatives, F'(x) = dF/dv * dv/dx - dF/du * du/dx = f(v) v' - f(u) u' So, for this problem, g(x) = ?[2x,3x] f(u) du g'(x) = f(3x)*3 - f(2x)*2 = 3...

**Intro to Trig**

you know that cos(x) = -1 at x = ?,3?,5?, ... So, we can check 7? = 21.99 9? = 28.27 Looks like C to me

**Math**

His gain on selling one bottle is -3, sohis gain on selling x bottles is just -3x Now plug in x=7 I get -21, not -12.

**Math**

not quite. Remember the chain rule dx/dy = 6y^3*(3y^4+2)^-1/2 Now you can see that (ii) is true Doing things implicitly, it's much clearer: x^2 = 3y^4+2 2x dx/dy = 12y^3 dx/dy = 6y^3/x

**Algebra Word Problem**

Europe: .36c Asia: 24 S America: c - .36c - 24 = .64c-24 Without some further information it is not possible to find a numeric value for c.

**physics**

1 km/hr = 20 m/s periods are not measured in m/s a = (-20m/s)/(4s) = -5 m/s^2 F = ma s = 1/2 at^2 ...

**Math**

correct.

**Math**

huh? It's just like the other one, assuming you know how to change mixed numbers to fractions. (5 7/8 km)(18/4 km) = (47/8 km)(9/2 km) = 423/16 km^2

**Math**

correct, but I'd reduce the fraction to 49/5 and the area is mm^2, not just mm

**Algebra I**

I used a suitable function for f(x) Since you did not specify one of your own, I felt free to pick a good one. I felt that since 42 is the answer to the Ultimate Question of Life, the Universe and Everything, it would be a good answer to your rather ill-posed question.

**Algebra I**

42

**math**

why not just use a regular pentagon, with apothem equal to the radius of the pizza?

**Maths**

To find the base area, 70.56N / (4900N/m^2) = 0.0144 m^2 = 144cm^2 v = 1/3 * 144 * 15 = 720 cm^3

**MAth**

2(x+y) = 240 2(1.1x+0.8y) = 240 now solve for x.

**Mathmatics**

AB has slope (1+7)/(-8+12) = 2 Now you have a point and a slope, so the line is y-5 = 2(x-2)

**Geometry**

?(100/81) = ?

**Maths**

SRS: 10*5*9 RSR: 5*10*4

**Maths**

b^0 = 1, so a=3 Now you know that 3b^2 = 12 I expect you can find b now ...

**Physics**

what is the critical angle, using Snell's law?

**Math, Ratio and Proportion**

the whole lawn is 3/3, or (3/2)*(2/3) so, it will take 3/2 * 52 = 78 minutes As a proportion, x/1 = 52/(2/3)

**Math**

3/(3/5) * 85 = $425

**Math**

nope 5000/500 * 3 = 30 or, keeping track of the units, you want to convert $$ to weeks: $5000 * 3weeks/$500 = 30 weeks

**AP math**

Yes. logs and exponents are inverse operations; one undoes the other: e^(-0.244t) = 3 -0.244t = ln3 t = ln3/-0.244 = -4.5

**exponent**

sorry - can't be done v^-7 * v^7 = v^(-7+7) = v^0 = 1

**math**

every time 8 days pass, you multiply by 1/2. So, after t days, 8 days have passed t/8 times, and the amount left is 5*(1/2)^(t/8) After 7 days, then, you have 5*0.5^(7/8) = 2.726mg 1mg is 1/5 of the starting amount. 1/5 is a bit less than 1/4 = (1/2)^2, so it should take a ...

**maths**

If the side parallel to the wall has length x meters, and the other side is y, we have x+2y = 2000 The area a is just a = xy = (2000-2y)y = 2000y - 2y^2 This is just a parabola, with its vertex (maximum value in this case) at y=500. So, x=1000, and the maximum area is 1000*500...

**precalculus**

exponential growth is always of the form p = a*e^(kt) that's why it's called exponential... The starting amount (at t=0) is a, since e^0 = 1.

**Trig**

Surely you mean a depth of 45m, not 45km! Draw a diagram. The length x of fishing line can be found via (45+1)/x = sin35°

**precalculus**

or, you can get all the exponents over a single value: 23(.95^t) = 15(1.11)^t (.95^t)/(1.11)^t = 15/23 (.95/1.11)^t = 15/23 t log(0.856) = log(0.652) t = log(0.652)/log(0.856) = 2.751 You can see that Reiny's value will agree, since subtracting logs lets you divide numbers...

**Math**

.85 * 40 = ?

**Algebra**

The turns are in the ratios 2:9 and 3:5=9:15 So, they are in the ratios 2:9:15 Now things are clearer, right?

**Algebra**

(15÷(3×5)+4)×8=40

**word problem , MATHS**

I see you'd rather keep posting this than actually work on it. He has 7 sections of material to use. (a) the dog run could be 3x15 (1x5 panels) or 6x9 (2x3 panels) (b) 3x18 or 6x15 or 9x12 Now, was that so hard?

**math**

so, what are those dimensions?

**math**

I'd say no, in general.

**Calculus**

Since the region is symmetric, we can just double the volume from x=0 to x=1. using shells of thickness dy, v = 2?[0,1] 2?rh dy where r=7-y and h=x=?y v = 4??[0,1] (7-y)?y dy = 256?/15 using discs of thickness dx, v = 2?[0,1] ?(R^2-r^2) dx where R=7-y and r=6 v = 2??[0,1] ((7-...

**Math - Calculus**

the x=0 boundary is redundant, since the parabola intersects the x-axis at 0 and 2. Anyway, the area is just a bunch of vertical strips of width dx and height y, so a = ?y dx = ?[0,2] 2x-x^2 = 4/3 Now, you want the line to divide R into two equal areas. y=cx intersects the ...

**math**

if A = LW then the new area A' = (2L)(2W) = 4LW = 4A so, 4A = 160

**Chemistry**

We are asked to prepare a 100 ml solution with the following 3 concentrations: 53.00 mg/ml sodium salicylate, 0.020 M FeCl3, and 0.050 M HCl. Then, we are going to take the absorbance at 520 nm. What is the best way to prepare the 100 ml solution ?

**Math**

No way to tell. If C has the same base area as D, then since its height is 1/2 as big, so is its volume. However, if C is similar to D, then all the dimensions are cut in half, so the volume is (1/2)^3 = 1/8 as much.

**maths**

huh? 2 boxes is 2*12 = 24 apples Think that's enough?

**maths**

168 = 3*56 = 2*56 + 56 so, there are 112 girls and 56 boys 56 = 7*8 I think the rest is not so hard, right?

**Algebra**

recall the discriminant of a quadratic: b^2-4ac Here, that is 1^2-4(-3)(-4) = 1+48 = 49 since it is positive, there are 2 real roots.

**reiny steve reiny damon!!! Damon help maths**

huh? surely you recall the distributive property (x-h) divides P(x), so P(x) = (x-h)*p(x) similarly, Q(x) = (x-h)*q(x) So, P-Q = (x-h)*p(x) - (x-h)*q(x) = (x-h)(p(x)-q(x)) so, x-h divides P-Q And above, I showed you what that is, and even factored it for you! So, those factors...

**reiny steve reiny damon!!! Damon help maths**

This just the same as showing that property with numbers. If a|p and a|q then a|p-q p = a*m and q=a*n p-q = a(m-n) so a|p-q for your polynomials, P-Q = (ax^3+x^2-15x-18)-(ax^3-14x-12) = x^2-x-6 = (x-3)(x+2) that should get you going, right?

**Check My Math Answers Please!**

#12 recall that the vertex of a quadratic (max height in this case) occurs when t = -b/2a = -32/-32 = 1

**Math**

1/3 x + 6 = 5/6 x 6 = 1/2 x x = 12 Check: the 1/3-full tank has 4 gallons. Adding 6 makes 10 gallons, or 5/6 of 12

**Math**

Surely not 3, since it does not appear in both top and bottom 8/15 * 5/16 = 8*5 / 15*16 = 40/240 Now, 2 can be factored out to give 20/120, but that is not fully simplified Same for 5 and 8, done individually. However, if 5 and 8 are both factored out, the resulting 1/6 is as ...

**algebra**

for the parallel line, you have a slope and a point, so use the point-slope form to get y+1 = 3(x+1) The perp. line has slope -1/3, so it is y+1 = -1/3 (x+1) Now massage those into slope-intercept form.

**MATH HELP ASAP PLEASE!!!!!**

You and "Naomi" should get together. can't read you attempts at fractions. Try something like 11 1/4 or 10 1/2 Consider using a regular pentagon with an apothem of 7", or the radius of the pizza. Look online to find the area of such a figure. It will be the ...

**math,**

Just write algebra for the conditions. Let w = municipal bonds x = CD's y = T-Bills z = Stocks Then the conditions are w <= (w+x+y+z)/5 x <= w, x <= y, x <= z x+y >= 3/10 (w+x+y+z) x+y >= 1.2(w+z) Rearrange things a bit, and you want to maximize p = .085w...

**Calculus**

find v(t), and then just take its derivative to find its maximum: v(t) = 3t^2-6t-9 v'(t) = 6t-6 clearly, v'=0 at t=1 However, since v' is just a parabola, its vertex is a minimum, not a maximum. That means that its maximum is either at t=0 or t=2. So, just compare ...

**maths**

You have mangled the question, but if I parse the meaning correctly, you need to check the amount of alcohol before and after mixing in the water: .60(80) = .40(80+x)

**maths**

the fine is just an arithmetic progression where the amounts (in thousands) are a = 4 d = 1 You want n where 4+1(n-1) <= 60

**vector calculus**

Take the cross product and the normal to both lines is <2,-2,1> The plane is then just 2(x-1)-2(y-5)+1(z-12) = 0 2x-2y+z = 4

**Math (Integrals)**

since ((2x-1)^3)/6 = 4/3 x^3 - 2x^2 + x - 1/6 the two solutions differ by 1/6 Since C is an arbitrary constant, its value does not matter.

**Algebra**

just add up the interest amounts: .05x + .10(30000-x) = 2500

**advance maths**

to find n, just solve n/2 (9+105) = 741 Now find d, using 9+(n-1)d = 105

**Statistics**

poke around here, and I think all will become clear: http://davidmlane.com/hyperstat/z_table.html

**math revise @Damon**

I think Damon went a bit overboard. The line has a slope, and it is zero. Not at all the same as having no slope at all. A vertical line's slope is undefined. Hmmm. Is that the same as not having a slope? Not sure...

**series maths steve reiny bob reiny steve!!!**

There are lots of sequences that fit those three numbers. Possibly the simplest is a quadratic, 10x^2-18x+16 So, p ? 10x^2-18x+16 = 2/3 p(5p^2-6p+13) k=1

**algebra**

I suspect you meant that they are draining. So, the two equations would be y = 75-7x and y = 115-3x for suitable domains of x.

**math**

You just need to remember the characteristics of parabolas. y^2?16y=12x?8^2 y^2-16y+64 = 12x-64+64 (y-8)^2 = 12x Now recall that for y^2 = 4px, we have vertex = (0,0) focus = (p,0) directrix is x = -p So, for your parabola, that means that it is shifted up 8, and p=3: vertex...

**Algebra (Please help!)**

I get 0.90 as the best match.

**Math**

12+5+3 = 20 5/20 = 1/4 12:5:3 * 1/4 = 3 : 5/4 : 3/4

**math**

Looks like 45 and 450 to me.

**Pre-Algebra**

what does the cylinder have to do with it? For the cone, v = ?/3 r^2 h = ?/3 (16)(3) = 16?

**Math**

the line joining the points has slope (-3+1)/(2+2) = -1/2 What do you know about the slopes of perpendicular lines?

**Pre-Algebra**

just divide salt/fresh: (3*10^20)/(6*10^18) = (3/6)*10^(20-18) = 0.5*10^2 = 50

**@Reiny gets the save**

Wow - I guess I mangled that one, eh?

**financial mathematics**

Where's the problem? Just plug your numbers into the formula, which I assume you have handy. A = Pr(1+r)^n/((1+r)^n-1) = 1000*0.16(1+.16/4)/((1+.16/4)^20-1) = 139.70

**math**

I assume your question is which two numbers have a product of -168 and a sum of 21? Well, check the factors of 168: 2 84 3 56 4 42 8 21 6 28 12 14 Now, which two differ by 21? None. So, your numbers are not rational. I suspect a typo. Algebraically, xy = -168 x+y = 21 x + (-...

**stat**

try poking around here, if you don't have a handy Z table: http://davidmlane.com/hyperstat/z_table.html

**Science**

Could be any of them, depending on how much of each you have. And what is H20? Molecules of hydrogen are just H2 :-)

**math**

218x - 36(30-x) = 5524

**maths geometry**

(a) ?r^2h = 450 a = 2?r^2+2?rh = 2?r^2+2?r(450/?r^2) = 2?r^2 + 900/r Now just find r when da/dr=0 (b) is similar... ?3/4 s^2h = 450 a = 2(?3/4 s^2) + 3sh = ?3/2 s^2 + 3s(1800/?3 s^2) = ?3/2 s^2 + 1800?3/s ...

**maths geometry**

I object!

**trigonometry and geometry**

2x+y = 2000 a = xy = x(2000-2x) da/dx = 2000-4x da/dx=0 when x=500 So, the maximum area is 500*1000 m^2 Note that this is achieved when the available fence is equally divided between lengths and widths.

**Help Please. Math**

No, your equation cuts the amount in half every time t increases by 1. Since the half-life is 5730 years, you want the amount to be cut by half only that often. So, A9t) = 600(0.5)^(t/5730) To do a sanity check on your answer, note that 40000 years is about 7 half-lives, so ...

**@Ms.Sue please help!!:(Math**

I = k/r^2 Now replace r with 5r, and you have I' = k/(5r)^2 = k/25r^2 = (1/25)(k/r^2) If r is replaced with nr, then I gets scaled down by a factor of 1/n^2

**Maths**

ever heard of google? It will provide lots of discussions on just what it is and what it is not.

**Maths**

It is defined that way. It makes possible the definition that n! = n(n-1)! for all integer n>0

**Calculus**

OK. Now you have ln(y-1) = -1/x + 1 y-1 = e^(-1/x + 1) y = e^(1 - 1/x) + 1

**math**

Maybe the sequence is squares of primes, since 4^2 and 6^2 are both missing.

**Mathematics-Geomeotry PLEASE HELPPPPPP!!!!!!!**

The prism is regular, meaning its bases are squares, of side s. Now you know that h=13 4sh = 624 So, s = 12 Now you can work out the rest.

**Mathematics-Geometry PLEASE HELP HELP!!!!!!**

the area of each base is just 10*3=30 Then you have four rectangles. Two that are 10x12 and two that are 6x12 Now just add up all those six areas.

**Maths**

4/3 the speed means 3/4 the time: 7.5 min.

**Differential Equations**

I know that sometimes finding the solution is tricky, but surely just taking the derivatives is easy, no? y = c1-cos(x+c2) y' = -sin(x+c2) y" = -cos(x+c2) (y")^2 + (y')^2 - 1 = 0 sin^2+cos^2 = 1, right?

**algebra**

Looks good to me.

**algebra**

a good place to check your work is http://calc101.com/webMathematica/long-divide.jsp

**math**

420(1+log4204)

**math**

4:1 = 64:16

**algebra**

1/2 of all that

**algebra**

4-2(-3)^k.....-8+8(-3)^k 1-(-3)^k......-2+4(-3)^k

**Calculus**

(a) x=2 2y^2-4y = 6 y^2-2y-3 = 0 (y-3)(y+1) = 0 So, the points are (2,3) and (2,-1) (b) xy^2-x^2y = 6 y^2 + 2xyy' - 2xy - x^2y' = 0 y'(2xy-x^2) = 2xy-y^2 y' = (2xy-y^2)/(2xy-x^2) (c) y'(2,3) = 3/8 y'(2,-1) = 5/8 check: http://www.wolframalpha.com/input...

**Math**

well, it will be 5/2 as long as NL