# 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.

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?

#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.

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

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...

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.

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 ...

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.

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.

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

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