Dear ninja02,
Although I am fully qualified to answer this question, I had a bit of
trouble making out some of your notation. I realize that transcribing
math equations from paper to screen is not an entirely straightforward
task. In order to make headway, I have used common sense and mathematical
intuition to resolve various inconsistencies and formatting peculiarities
as best as I could.
If any of my assumptions are incorrect, and in fact you meant something
else in one or more of the equations, I urge you to post a clarification
request with full details, and I will respond promptly with updated
solutions wherever necessary. It is very important to me that I get your
question exactly right.
So here goes.
Exercise 1.
I assume that F, L, and C are variables in this equation, which you want
to solve for F. I shall use the asterisk, "*", to denote multiplication,
and the form "sqrt(...)" for the square root operation. Furthermore, I
shall use parentheses to disambiguate my notation as much as possible,
though you will find that you can dispense with many of these in
handwritten form.
2pi*F*L = 1/(2pi*F*C) ; given
2pi*(F^2)*L = 1/(2pi*C) ; multiply both sides by F
F^2 = 1/((4pi^2)*L*C) ; divide both sides by 2pi*L
F = sqrt(1/((4pi^2)*L*C)) ; take the square root of both sides
F = (1/(2pi))*sqrt(1/(L*C)) ; remove 1/(4pi^2) from under the root
Exercise 2.
T = sqrt(2pi*L*C) ; given
T^2 = 2pi*L*C ; square both sides
T^2/(2pi*L) = C ; divide both sides by 2pi*L
C = T^2/(2pi*L) ; transpose
Exercise 3.
I assume that the entire fraction is raised to the power of 3/2, and
that the task is to ensure that all powers are positive.
( 0.5e^(-0.2t) / e^(-0.6t) )^(3/2) ; given
( 1/2 * e^(-0.2t)/e^(-0.6t) )^(3/2) ; render 0.5 as a fraction
( 1/2 * e^(-0.2t - (-0.6t)) )^(3/2) ; subtract powers of e
( (e^0.4t)/2 )^(3/2) ; simplify the exponential
(e^0.4t)^(3/2) / 2^(3/2) ; bring the power inside
e^(0.4t*3/2) / sqrt(2^3) ; multiply out the powers
e^(0.6t) / (2*sqrt(2)) ; simplify exponential and root
Exercise 4.
x^2 - 4x = 5 ; given
x^2 - 4x - 5 = 0 ; subtract 5 from both sides
x^2 - 5x + x - 5 = 0 ; complete the square
(x - 5)(x + 1) = 0 ; factorize
We conclude that the solutions are 5 and -1.
Exercise 5.
I have omitted your "RAD" notation, as it is not usual in
trigonometry. The convention is that degrees are denoted explicitly,
whereas an angle expressed simply as a number is understood to be in
radians. Note that I have carried out all calculations with two decimal
digits of precision.
Part A.
6*sin(x+1.2) = -4 ; given
sin(x+1.2) = -4/6 ; divide both sides by 6
sin(x+1.2) = -2/3 ; simplify the fraction
x+1.2 = asin(-2/3) ; take the inverse sine of both sides
x+1.2 = (-0.73) or (pi+0.73) ; two angles are possible
x = (-1.93) or (pi-0.47) ; subtract 1.2 from both sides
x = (2pi-1.93) or (pi-0.47) ; add 2pi to the first angle
x = 4.35 or 2.67 ; calculate
Part B.
3*cos(a-pi) = -1 ; given
cos(a-pi) = -1/3 ; divide both sides by 3
a-pi = (1.91) or (-1.91) ; two angles are possible
a = (pi+1.91) or (pi-1.91) ; add pi to both sides
a = 5.05 or 1.23 ; calculate
And that's the lot!
Once again, I urge you to post a clarification request immediately if it turns
out that I've misconstrued any of the exercises. Also, if you have any other
questions or concerns, please post a clarification request so that I have a
chance to meet your needs before you assign a rating.
Cheers,
leapinglizard |
