# Posts by steve

Total # Posts: 50,558

math
since all the sides are equal, the altitude is 10 sin60° = 5?3 To see this, draw the rhombus ABCD and drop an altitude from D to AB. Now you have a right triangle with hypotenuse=10 and base angle=60°

math
Since the diagonals bisect the vertex angles, if ?K is obtuse, ?PKE cannot be just 16°. Anyway, to solve this just remember that two consecutive angles of a rhombus are supplementary, so if half of one of them is 16°, half of the next one is 90-16=74° The diagonals...

Pre calc

PreCalculus
draw a diagram. It is clear that (a) tan? = 20t/50 (b) solve when t=3 Since tan2??2tan?, (c) is clearly false.

Math
If the center is at O, then the slope of OP = 3/4 So, the tangent line, which is perpendicular to the radius OP has slope -4/3, and its equation is thus y-3 = -4/3 (x-4) The line's x-intercept is at Q=(7,0) Due to symmetry, the other tangent line through Q touches the ...

Math
I'll do (c) and the others work the same way. You just complete the squares: x^2 +y^2 +2x?8y = ?8 x^2+2x + y^2-8y = -8 x^2+2x+1 + y^2-8y+16 = -8+1+16 (x+1)^2 + (y-4)^2 = 3^2 Now you can just read off the center and radius.

Math
as in your other post, complete the squares and all will be clear.

limit maths sir steve or damon or reiny help
(e2x)1/x < (x+ex+e2x)1/x < (2e2x)1/x e2 < (x+ex+e2x)1/x < 21/xe2 Now take the limits, and since 21/x-> 1, e2 < (x+ex+e2x)1/x < e2 so (x+ex+e2x)1/x = e2 Or, you can take the log of the limit, and then use l'Hospital's Rule

Physical Science
A current of 6.0 amps passes through a motor that has a resistance of 5.0 ohms. calculate the power. my answer is 180 watts

Calc
#1 The LS can be written as         sinx(cosx-sinx) ---------------------------------- sinx(cosx+sinx)(cosx-sinx) I think it's clear what comes next ... For #2, on the LS, multiply top and bottom by (1+sinx) and I think it will come ...

Pre calc
remember your double-angle formulas: cos(2?) = 4 ? 3 cos(?) 2cos^2(?)-1 = 4 - 3cos? 2cos^2(?)+3cos? - 5 = 0 (2cos?-1)(cos?+5) = 0 cos? = -5 has no solution cos? = 1/2 has solutions at ? = ?/3 and 5?/3 and other basic trig identities: cos(2?)cos? = sin(2?)sin? (2cos^2(?)-1)cos...

math
find the volume of 2 pyramids. 2* 1/3 Bh multiply by gold cos/mm^3 find area of 8 triangles. 8* 1/2 Bh multiply by paint cost/mm^2

futher maths
z^2 = 10^2 + 8^2 - 2*10*8 cos120°

Precalculus
Take a look here for an example: y = (x+2)^2 * (x-1)^3 http://www.wolframalpha.com/input/?i=(x%2B2)%5E2+*+(x-1)%5E3

Precalculus
if a factor (x-h) occurs n times, it has a multiplicity of n. Looking at the graph, if a root has odd multiplicity, the graph crosses the x-axis there. Think of y=x^3 If the multiplicity is even, the graph just touches the x-axis and turns back. Think of y=x^2

Algebra
y=4x and y=6x go through the origin (0,0) 10+4x starts off 10 units higher. 6+6x starts off 6 units higher. The y-intercepts are 10 and 6. The x-intercepts are -5/2 and -1 (2,18) is where the two lines intersect. It is the solution to both equations.

Geometry
take a look at the Angle Bisector Theorem AX/BX = CA/CB

maths sir steve help me reiny
The curve is just a parabola: x = y^2/4a So, the area is just A = ?[1,2] y(t) dx(t) dx = 2at dt A = ?[1,2] 2at * 2at dt = ?[1,2] 4a^2t^2 dt = 4/3 a^2 t^3 [1,2] = 4/3 a^2 (8-1) = 28/3 a^2 To check, express y as a function of x: A = ?[a,4a] y dx = ?[a,4a] 2?(ax) dx = 2?a (2/3 x...

mathematics
x = ky+m 7 = 5k+m 8 = 7k+m so, k = 1/2 and m = 9/2 ...

science
less dense

Calculus
Well, there are two regions, on the intervals [-3,0] and [0,1] Finding the areas is pretty straightforward, right?

physics
v = ?(20/2)^2 * 0.05 = 5? cm^3

Math
I guess that'd depend on how many yellow balloons there were at the start.

math
slide depends on the flips

math
the altitude of the triangle is the distance between the lines. A = 1/2 bh b is constant, h is constant.

math
If the diagonals AC ? BD then the figure is a rhombus. Since the area of a rhombus is the one-half product of the diagonals, (29/2)*BD/2 = 58 Not so hard now, eh?

MAth
500x-100(28-x)=9100

Trigonometry
nope. Although you do know that the period is 2?, so it crosses the x-axis at intervals of ?, with cos(0) = 1 cos(?/2) = 0 cos(?) = -1 cos(3?/2) = 0 cos(2?) = 1 Then sketch a smooth curve using those points. If you want other points, you can use known values at ?/6, ?/4, ?/3 ...

Pre calc
2sin ?/3 + ?3 = 0 sin ?/3 = -?3/2 ?/3 = 4?/3 or 5?/3 + 2n? ...

Math
4x-3y = -4 3x-2y = -4 8x-6y = -8 9x-6y = -12 Now subtract. The y disappears and you get -x = 4 x = -4 now use that to get y.

Math
a triangle of side 24/3 = 8 a square of side 24/4 = 6 and so on. If you want to include odd shapes, than just use your imagination...

Calculus
It is wrong. Your equation uses a rate of 2.1%. a .21% growth rate means 6.8 * 1.0021^(t-1988) You can use a base of e, but that would be ln 1.21 = 0.0021 6.8 e^(0.0021(t-1988)) or, if t is defined as the number of years since 1988, 6.8*1.0021^t or 6.8 e^(0.0021t)

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...