Questions

[mathmajor]

Here are some questions on the most recent meet.

12. Consider line m which is coplanar to and equidistand from (6,4) and (2,-8). Find the y-intercept of line m.

34-35. A mass on the end of a spring is oscillating vertically and its position can be modeled by the equation y(t) = 4cos12t, 0 <t < pi

34. Find the period of oscillation

35. Find the speed of the mass when the acceleration equals 24.

43. Find the direction angle in radians for u=4i-8j

44. TO write the partial fraction decomposition of (2x-3)/(x^3+10x) begin with A/x + (Bx+C)/(x^2+10). THen A + B + C =

58. Find the probability that Jo, an 88% free throw shooter, will make at least 8 of his 10 free throws in a game.

60. Find the arc length of the curve on the given interval. x=4t^2, y=6t, 0 <t < 4

[Shawn007]

A few to add to the list:
37. To evaluate integration(sin^2(x)xdx), use the substitution x= ???
51-52. Consider an ellipse with vertices (-3,10) and (-3,0) and foci (-3,8) and (-3,2)
51. Find the area bounded by the graph of the ellipse.
52. Find the approximiate perimeter of the ellipse.
57. To evaluate integration(x^2ln(x)dx) use integration by parts and let u=______
59. The series defined by summation of(1/(n*ln(n)) as n goes from 2 to infinity) diverges/converges to what number?
Thanks I am going to try to work some of those on the post above this.
Thanks for your help,
Shawn

[Shawn007]

My solutions:
12.
d1=d2
(x-6)^2+(y-4)^2=(x-2)^2+(y+8)^2
x^2-12x+36+y^2-8y+16=x^2-4x+4+y^2+16y+64
-12x+36-8y+16=-4x+4+16y+64
-24y=8x+16
y=-8/24x+-2/3
34.
Normal period of cosine= 2(pi)
4cos(12t)
New period = 2(pi)/12
New period=pi/6
35.
Position: y(t)=4cos12t, on the interval of t (0,pi)
Velocity/Speed is the first derivative of position. Acceleration is the second.
y'(t)=-48sin(12t)
y''(t)=-576cos(12t)
24=-576cos(12t)
t= 1.705169
y'(1.075169)= -48
That is all I know how to do.

[bradp]

43. Find the direction angle in radians for u=4i-8j

arctan(-8/4)?

44. TO write the partial fraction decomposition of (2x-3)/(x^3+10x) begin with A/x + (Bx+C)/(x^2+10). THen A + B + C =

(2x-3)/(x^3+10x) = A/x + (Bx+C)/(x^2+10)
multiply everything by the original denominator
2x-3=A(x^2+10)+(Bx+C)(x)
one equation, too many unknowns - so pick a convenient x so that some unknowns go away
So make either (Bx+C)(x) = 0 or A(x^2+10) = 0
x=0 is the obvious choice
2(0)-3=A(0^2+10)+(B*0+C)(0)
-3=10A
A=-10/3

Now, you can plug in any other x values to find B and C. Or you can pick a convenient x that will give you B+C in the equation. Since you only need to find A+B+C, knowing A and B+C is sufficient. You don't have to find their actual values, just their sum.


58. Find the probability that Jo, an 88% free throw shooter, will make at least 8 of his 10 free throws in a game.

Look up binomial probability distribution

60. Find the arc length of the curve on the given interval. x=4t^2, y=6t, 0 <t < 4

Consider a small piece of the curve ds (d as in differential and s as in arc length)

Pythagorean theorem (assume that the curve is linear over that extremely small interval)
(ds)^2=(dx)^2+(dy)^2
divide everything by (dt)^2. Note that these are not second derivatives, just squared differentials

ds^2/dt^2=dx^2/dt^2 + dy^2/dt^2

ds=sqrt((dx/dt)^2 + (dy/dt)^2)dt

S = integral of sqrt( (dx/dt)^2 + (dy/dt)^2 ) with respect to t

so

S = integral of sqrt( (8t)^2 +(6)^2 ) from t = 0 to t=4

[bradp]

51-52. Consider an ellipse with vertices (-3,10) and (-3,0) and foci (-3,8) and (-3,2)

51. Find the area bounded by the graph of the ellipse.
Shift the ellipse's center to (0,0) first to make it easier. Find the equation for the ellipse:
a = (10-0)/2 = 5
c = (8-2)/2=3
b^2=a^2-c^2 --> b=4

the equation of the ellipse is x^2/16+y^2/25=1
or x=plus/minus 4sqrt(1-y^2/25)
integrate the plus version of that equation with respect to y from y = 0 to y = 10 (or -5 to 5 since we put the center at (0,0) ). Multiply by two to get the total area. Drawing a graph could help you visualize this.

[RussiaPSJA]

[kvalle]

57. Whenever you are asked to use integration by parts, it is best to use ln(x) as u. It is a lot easier than antidifferentiating ln(x)

[kvalle]

[bradp]

"Actually scratch that, I don't know why you would ever do that."

So... where did pi*a*b come from in the first place? Of course I would never do that integration on a UIL test, but sometimes its good to know where things come from. And you still have to integrate to find the perimeter.

Plus, those questions look like they're from some calculus homework, not a UIL test.

[AuSmith]