Friday, May 31, 2013

Using double angle formula find the exact value

, given secθ=4,3pi/2<θ<2pi 
sin2θ= 
cos2θ= 
tan2θ=


sec%28theta%29=4


cos%28theta%29=1%2F4 

3pi%2F2%3Ctheta%3C2pi 

i.e. we are in quadrant IV 

means... 

sin%28theta%29=-sqrt%2815%29%2F4 
 


cos2%2Atheta=2cos%5E2%28theta%29-1=2%281%2F4%29-1=-1%2F2 


We forgo using tangent identities since we don't need them 

Wednesday, May 29, 2013

Solve: 16x^(3)-100x=0


%0D%0A16x%5E%283%29-100x=0%0D%0A



%0D%0Ax%28%284x%29%5E2-10%5E2%29=0%0D%0A

x=0
4x=10, x=5/2
4x=-10, x=-5/2


:)

how much pure acid should be mixed with 3 gallons of 20% acid solution to get a 40% acid solution?


unknown amt: x
20% acid: 3(0.2)
new mixture: (x+3)(0.4)
%0D%0Ax%2B%283%29%280.2%29=%28x%2B3%29%280.4%29%0D%0A

x=1




:)

A purse contains 15 coins worth $1.35. If there are only dimes and nickels in the purse, how many coins of each kind are there?


informally this problem only requires a working knowledge of counting money
12 dimes: $1.20
3 nickels:$0.15
buck thirty five!

formally:
coins: n+d=15
cash value: 0.05n+0.10d=$1.35 (which we multiply by 20 for convenience)
n+2d=27
then
(n+2d)-(n+d)=27-15

d=12
so n=3



:)

Verify the basic identity. What is the domain of validity? cotθ = cosθcscθ


%0D%0Acot%28theta%29=cos%28theta%29%2Fsin%28theta%29=cos%28theta%29csc%28theta%29%0D%0A



and it has the domain


%0D%0A0%3C=theta%3C%2Bpi%0D%0A


:)

Simplify: -1 + ((x - b)( x - c ))/((a-b)(a-c)) +( (x - a)(x - b))/((c-a)(c-b)) +( (x - a)(x - c))/((b-a)(b-c))

The simplification is relatively lengthy so I will outline it: 
First off we set aside the -1 term and simplify the rational expression part. 

the lcd is 

(a-b)(a-c)(b-c) 

After some labor we arrive at these results: 
x^2 terms: 0 
ax terms: abx-acx 
bx terms: bcx-abx 
cx terms: acx-bcx 
bc terms: bc(b-c) 
ab terms: ab(a-b) 
ac terms: -ac(a-c) 

After collecting like terms we find that all x-terms add out (cancel each other to get a sum of 0) 

Leaving us with 

ab(a-b)-ac(a-c)+bc(b-c) 

which is equal to (a-b)(a-c)(b-c), the denominator! 

Therefore our rational expression simplifies to 1 so that the entire expression evaluates to 0 










:) 

Sunday, May 26, 2013

Evaluate: (4/5+1/10)^(2)/(3/2)^(2)

what is the inverse of f(x)= 2x-3

y=2x-3 


y%2B3=2x 



%28y%2B3%29%2F2=x 


so... 

f%5E%28-1%29%28x%29=%28x%2B3%29%2F2

Solve 7+5|c|≤1-3|c|

%0D%0A7%2B5abs%28c%29%3C=1-3abs%28c%29%0D%0A 


%0D%0A5abs%28c%29%2B3abs%28c%29%3C=1-7%0D%0A 

%0D%0A8abs%28c%29%3C=-6%0D%0A 

W can stop immediately because the absolute value can never be negative so this inequality has no solution 

:) 

what is the value of x in 6/x = 7/(x+3)



%0D%0A6%2Fx+=+7%2F%28x%2B3%29%0D%0A 
then... 



We cross multiply because this is a proportion:
%28x%2B3%296=7x 




%0D%0A6x%2B18=7x%0D%0A 



x=18

Solve: abs(2x-5)<=8

abs%282x-5%29%3C=+8 


which means 
%0D%0A-8%3C=2x-5%3C=8%0D%0A 


Now we simplify... 



%0D%0A-8%2B5%3C=2x-5%2B5%3C=8%2B5%0D%0A 


%0D%0A-3%2F2%3C=%282x%29%2F2%3C=13%2F2%0D%0A 




%0D%0A-3%2F2%3C=x%3C=13%2F2%0D%0A 


:) 

Find all points on the y-axis that are 6 units from (4, -3).

We'll use this form: 

%0D%0Ad%5E2=%28x%5B2%5D-x%5B1%5D%29%5E2%2B%28y%5B2%5D-y%5B1%5D%29%5E2%0D%0A 

%0D%0A6%5E2=%284-0%29%5E2%2B%28-3-y%5B1%5D%29%5E2%0D%0A 



%0D%0A%28-3-y%5B1%5D%29%5E2=36-16%0D%0A 


%0D%0Ay%5B1%5D=sqrt%2810%29-3%0D%0A 
or 

%0D%0Ay%5B1%5D=-sqrt%2810%29-3%0D%0A 



:) 

Write xy=12 into polar form

%0D%0Axy=12%0D%0A 

r%2Acos%28theta%29%2Ar%2Asin%28theta%29=12 


r%5E2%2Asin%282%2Atheta%29%2F2=12 


%0D%0Ar%5E2=24csc%282%2Atheta%29%0D%0A 



:)

Solve |2x|<=|x-3|

We start by finding points from which we will find the boundaries for the intervals 

2x=x-3 or 2x=-x+3
x=-3 or x=1 

Next we test all intervals 

On x<-3 we test x=-4 and see 8<7 (false) 

On -3 On x>1 we test x=2 and see 4<1 (false)

Our solution interval: [-3,1]


:)

Find the slope of the line perpendicular to the line that passes through the following points (4,11) and (-2, 1)

%0D%0Am=%2811-1%29%2F%284-%28-2%29%29=10%2F6=5%2F3%0D%0A 

Now recall that 

%0D%0Am%5Bperpendicular%5D=-1%2Fm=-3%2F5%0D%0A 


:)

find all sets of three consecutive positive even integers with a sum no greater than 36

the set of solutions is infinite: 

%0D%0A%28n%29+%2B+%28n%2B1%29+%2B+%28n%2B2%29%3C=36%0D%0A 

leads to n=11, but n must be even so n=10 

Our list of consecutive even integers with a sum less than or equal to 36 ends with {10, 12, 14} but it has no beginning. 

:)