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x^2/36-y^2/16=1
This is an equation of a hyperbola with horizontal transverse axis.
Its standard form: (x-h)^2/a^2-(y-k)^2/b^2=1, (h,k)=(x,y) coordinates of the center
For given equation:x^2/36-y^2/16=1
center: (0,0)
..
a^2=36
a=√36=6
vertices: (0±a,0)=(0±6,0)=(-6,0) and (6,0)
..
b^2=16
b=√16=4
conjugate points: (0,0±b)=(0,0±4)=(0-4) and (0,4)
..
c^2=a^2+b^2=36+16=52
c=√52≈7.2
foci: (0±c,0)=(0±7.2,0)=(-7.2,0) and (7.2,0)
..
slope of asymptotes for hyperbolas with horizontal transverse axis=±b/a=±4/6=±2/3
asymptotes are straight lines that intersect at the center.
Equation for asymptote with negative slope:
y=-2x/3+b
since y-intercept is at 0, b=0
equation of asymptote: y=-2x/3
..
Equation for asymptote with positive slope:
y=2x/3+b
since y-intercept is at 0, b=0
equation of asymptote: y=2x/3

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x^2/25+y^2/16=1
What is the foci?
This is an equation of an ellipse with horizontal major axis
Its standard form: (x-h)^2/a^2+(y-k)^2/b^2=1, (a>b), (h,k)=(x,y) coordinates of the center
For given equation:
center: (0,0)
a^2=25
b^2=16
c^2=a^2-b^2=25-16=9
c=√9=3
foci: (0±c,0)=(0±3,0)=(-3,0) and (3,0)
- -
Given 36x^2+9y^2-324=0
36x^2+9y^2=324
divide by 324
x^2/9+y^2/36=1
This is an ellipse with vertical major axis
center: (0,0)
..
a^2=36
a=√36=6
vertices:(0,0±a)=(0,0±6)=(0,-6) and (0,6)
..
b^2=9
b=3
co-vertices:(0±b,0)=(0±3,0)=(-3,0) and (3,0)
..
c^2=a^2-b^2=36-9=25
c=√25=5
foci:(0,0±c)=(0,0±5)=(0,-5) and (0,5)

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3¼/2 = B/two-thirds
3 and 1/4=13/4
rewrite equation:
(13/4)/2=B/(2/3)
cross-multiply
2B=(2/3)(13/4=26/12
divide by 2
B=26/24=13/12
If an exact answer is wanted, you must leave it in this fraction form.
If a decimal answer is wanted, you will get 13/12=1.08333...
so, you would probably round it off depending on how many decimal places are wanted.
one decimal place=1.1
two decimal places=1.08
three decimal places=1.083
etc.
hope this helps

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a person drives from town A to Town B at the rate of 40mph and then flies back at the rate of 120mph. if the total traveling time is 11 hours, how far is it in miles from town A to town B?
**
let x=travel time from town A to town B
11-x=travel time on return trip
Distance=travel time*speed
..
40x=120(11-x)
40x=1320-120x
40x+120x=1320
160x=1320
x=8.25 hrs
11-x=2.75 hrs
distance from town A to town B=8.25*40=330 miles

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verifying identites
csc(x)*cos^2(x)+sin(x)=csc(x)
**
Start with left side:
csc(x)*cos^2(x)+sin(x)
=csc(x)(1-sin^2x)+sin(x)
=cscx-cscxsin^2x+sinx
=cscx-(1/sinx)*sin^2x+sinx
=cscx-sinx+sinx
=cscx
verified:
left side=right side

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Find all solutions to the equation.
csc theta-sqroot of 2=0
cscx-√2=0
cscx=√2
1/sinx=√2
sinx=1/√2
x=π/4+2πn and 3π/4+2πn, n=integer

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find all the solution of the equation in the interval (0,2pi)
tan(x+pi)+2 sin(x+pi)=0
tan(x+π)/sin(x+π)=-2
[sin(x+π)/cos(x+π)]/sin(x+π)=-2
sin(x+π) cancels out
1/cos(x+π)=-2
cos(x+π)=-1/2
x+π=2π/3 and 4π/3 (in quadrants II and III where cos<0)
x+π=2π/3
x=2π/3-π=-π/3
and
x+π=4π/3
x=4π/3-π=π/3
ans:
x=-π/3 and π/3

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Find all the solution of the equation in the interval (0,2pi)
sec^2x-1=0
tan^2x=1
tan x=±√1=±1
x=π/4, 3π/4, 5π/4, and 7π/4
..
sinx-2=cosx-2
sinx=cosx
sinx/cosx=1
tanx=1
x=π/4 and 5π/4 in quadrants I and III where tan>0

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interval:(0 to π)
solve the equation
√3 cscx-2= 0
cscx=2/√3
1/cosx=2/√3
cosx=√3/2
x=π/6
..
Solve the equation
tanx+ √3 = 0
tanx=-√3
x=2π/3 (reference angle in quadrant II where tan<0)

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using periodic function, find the value of sin 17pie
-
3
convert 144degrees int radians
**
17π/3=5π+(2/3)π which gives you an angle of 5π/3 in standard position or a reference angle of π/3 in quadrant III.
sin(17π/3)=sin(π/3) =-√3/2 (in quadrant III where sin<0)
..
144º=(144/180)π=0.8π radians

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How do you find the equation, vertex, focus, directrix, and latus rectum of the following:
-14x+2y^2-8y=20
complete the square
2(y^2-4y+4)=20+14x+8
2(y-2)^2=14x+28
divide by 2
(y-2)^2=7x+14
(y-2)^2=7(x+2)
This is an equation of a parabola that opens rightwards.
Its standard form: (y-k)^2=4px, (h,k)=(x,y) coordinates of the vertex
For given equation:(y-2)^2=7(x+2)
vertex:(-2,2)
axis of symmetry: y=2
4p=7
p=7/4
Focus: (-2+p,2)=(-2+7/4,2)=(-1/4,2) (p distance to the right of the vertex on the axis of symmetry)
Directrix: x=(-2-p)=(-2-7/4)=-15/4 (p distance to the left of the vertex on the axis of symmetry)
latus rectum:
length of latus rectum=4p=7
2p=7/2
end points: (-1/4,2±2p)
=(-1/4,2±7/2)
=(-1/4,-1.5) and (-1/4,5.5)

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A point on the rim of a wheel has a linear speed of 18 cm/s. If the radius of the wheel is 20 cm, what is the angular speed of the wheel in radians per second?
**
cm=centimeter, rev=revoltion, r=radius
18cm/sec *rev/2πr*2π radian/rev
cm, rev, 2π cancel out, leaving
18/r radians/sec
angular speed of wheel=18/20=9/10 radians/sec

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how do you find the vertices, co-vertices, foci, and asymptotes of:
9(x+2)^2 -16(y-1)^2=-144
divide by -144
-(x+2)^2/16+(y-1)^2/9=1
(y-1)^2/9-(x+2)^2/16=1
This is an equation of a hyperbola with vertical transverse axis.
Its standard form: (y-k)^2/a^2-(x-h)^2/b^2=1, (h,k)=(x,y) coordinates of center
For given equation: (y-1)^2/9-(x+2)^2/16=1
center: (-2,1)
a^2=9
a=3
vertices:(-2,1±a)=(-2,1±3)=(-2,-2) and (-2,4)
..
b^2=16
b=4
co-vertices:(-2±b,1)=(-2±4,1)=(-6,1) and (2,1)
..
c^2=a^2+b^2=9+16=25
c=√25=5
foci::(-2,1±c)=(-2,1±5)=(-2,-4) and (-2,6)
..
Asymptotes are straight lines that intersect at the center
slopes of asymptotes=±a/b=±3/4
..
Equation of asymptote with negative slope
y=mx+b, m=slope, b=intercept
y=-3x/4+b
Solve for b using coordinates of center
1=-3*-2/4+b
b=1-6/4=-2/4=-1/2
Equation: y=-3x/4-1/2
..
Equation of asymptote with positive slope
y=mx+b, m=slope, b=intercept
y=3x/4+b
Solve for b using coordinates of center
1=3*-2/4+b
b=1+6/4=10/4=5/2
Equation: y=3x/4+5/2

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draw asymptotes (if any), to graph the rational function, plot at least 2 points on each piece of the graph x^2+5x+5/x+2
**
I don't have the means to plot a graph for you, but I can provide information with which you can plot the curve yourself.
When degree of numerator is one unit higher than that of the denominator, you will have slant or oblique asymptotes, as in this case.
To find equation of slant asymptote, divide numerator by denominator by long division or synthetic division. The quotient is the equation of the slant asymptote.
(x^2+5x+5)/(x+2)=(x+3)+remainder=1
equation of slant asymptote: y=x+3
..
To find vertical asymptote, set denominator=0, then solve for x
x+2=0
x=-2=vertical asymptote
..
y-intercept
set x=0,then solve for y
f(0)=(x^2+5x+5)/(x+2)=5/2
y-intercept=5/2
..
x-intercepts
set y=0
x^2+5x+5=0
solve by quadratic formula:
x=-3.62 and -1.38
..
number line
<....-.....-3.62...+...-2...-...-1.38...+....>
see graph below:

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use the sum and difference formulas to solve the equation on [0,2pi):
cos(x+5pi/6)-cos(x-5pi/6)=1
**
cos(x+5pi/6)-(cos(x-5pi/6)=1
cosxcos5π/6-sinxsin5π/6-(cosxcos5π/6+sinxsin5π/6)=1
cosxcos5π/6-sinxsin5π/6-cosxcos5π/6-sinxsin5π/6)=1
-2sinxsin5π/6=1
sin(5π/6)=1/2 in quadrant II
sinx=1/-2sin5π/6=1/(-2*(1/2))=1/-1=-1
x=3π/2

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Use difference of two angles identity to find the exact value of sin 15 degrees.
sin 15º=sin(45-30)
=(sin45cos30-cos45sin30)
=√2/2*√3/2-√2/2*1/2
=√6/4-√2/4
=(√6-√2)/4

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verify the identity
tanx(1-sinx^2x)=1/2sin2x
**
Start with left side
tanx(1-sinx^2x)
=(sinx/cosx)(cos^2x)
=sinxcosx
=(1/2)sin2x
verified:
left side=right side

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Two cars are at a distance of 600 km. These are moving in opposite direction and if there is a difference of 6 km/h in their speeds and after 4 hours distance between them is 123km then find the speed of each car
**
let x=speed of slower car
x+6=speed of faster car
distance traveled=600-123=477 km
travel time=4 hrs
Distance=travel time*speed
4x+4(x+6)=477
4x+4x+24=477
8x=477-24=453
x=453/8=56.625
x+6=62.625
ans:
speed of slower car=56.625 km/hr
speed of faster car=62.625 km/hr

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Jack, Kay, and Lynn deliver advertising flyers in a small town. If each person works alone, it takes Jack 4 hours to deliver all the flyers, and it takes Lynn 1 hour longer than it takes Kay. Working together, they can deliver all the flyers in 40% of the time it takes Kay working alone. How long does it take Kay to deliver all the flyers alone?
**
let x=hrs Kay takes to do the job alone
1/x= Kay's work rate
x+1=hrs Lynn takes to do the job alone
1/(x+1)=Lynn's work rate
4=hrs Jack takes to do the job alone(given)
1/4 =Jack's work rate
.4x=hrs to complete the job working together
..
.4x/x+.4x/(x+1)+.4x/4=100% of the job
.4+.4x/(x+1)+.x=1
multiply by 10
4+4x/(x+1)+x=10
LCD:(x+1)
4x+4+4x+x^2+x=10x+10
x^2-x-6=0
(x-3)(x+2)=0
x=-2 (reject, x>0)
or
x=3
ans:
hrs Kay takes to do the job alone=3

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-x/7+4≥3x
multiply both sides by 7
-x+28≥21x
-22x≥-28
divide both side by -22 and reverse inequality sign
x≤28/22=14/11
solution: (-∞,14/11]

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Average age of 4 sons of a family is 12 years. Average age of sons together with their parents is 28 years. If father is older than mother by 6 years . Find the age of mother?
**
let x=age of mother
x+6=age of father
..
Total age of 4 sons/4=12
Total age of 4 sons=48
..
(Total age of 4 sons+father's age+mother's age)/6=28
(48+x+6+x)/6=28
48+x+6+x=168
2x=168-54=114
x=114/2
x=57
ans:
age of mother=57

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Find the solution(s) on the interval 0 ≤ theta<2(pi)
2cos(theta)+ √3 = 0
2cosx=-√3
cosx=-√3/2
x=5π/6 and 7π/6 (in quadrants II and III where cos<0)

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I have a question, what is the smallest positive real number x such that sin(2x+1)=1? Please give me a full explanation.
**
When sin(x)=1
x=π/2 radians
so, when sin(2x+1)=1
2x+1=π/2 radians
4x+2=π
4x=π-2
x=(π-2)/4

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log5 10+log5 2-log5 4
=log5[10*2/4]

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Write the ordered pair for f(-5) = 8 and identify the x- and y-values.
ordered pair=(-5,8)
x=-5
y=8

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Please Help me Condense 3 Log2 X-3Log2 2
3log2(x)-3log2(2)
=log2[x^3/2^3]
=log2[(x/2)^3]

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The word problem is For an object dropped from a height ho above the ground,its height after t seconds is given by h(t)=-16t^2+ho,where H is measured in feet. A ball is dropped from the top 96 foot tall building. How long will it take to fall to the ground? How long will it take to fall half of the distance to ground level?
**
h(t)=-16t^2+ho
For given problem, ho=96
h(t)=-16t^2+96
..
When object falls to the ground, h(t)=0
0=-16t^2+96
16t^2=96
t^2=96/16=6
t=√6
..
When object falls half the distance to the ground, h(t)=48
48=-16t^2+96
16t^2=48
t^2=48/16=3
t=√3
ans:
When object falls to the ground, it will take√6 seconds
When object falls half the distance to the ground,it will take √3 seconds

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you have sold 36 units of product Y. you are currently at 80% of you goal. how many more units need to be sold before you reach your goal?
**
let x=total number of units need to be sold to reach your goal
36/x=.8
x=36/.8=45
ans:
additional units need to be sold to reach your goal=45-36=9

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A company that manufactures bicycles has a fixed cost of $90000. It costs $110 to produce each bicycle. The total cost for the company is the sum of its fixed costs and variable costs. First write the total cost, , as a function of the number of bicycles produced
**
The formula to be used is that of a straight line.
Its standard form: y=mx+b, m=slope, b=y-intercept
For given problem:
m=110/bicycle
b=y-intercept=90000
x=number of bicycles produced
Equation for total cost: y=$110x+$90000

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If a tire rotates at 400 revolutions per minute when the car is travelling 72km/h, what is the circumference of the tire?
**
Use quantitative analysis method to solve:
let c=circumference
..
72km/hr*min/400 rev*hr/60min*rev/c
hr, min, rev, cancel out
=72000m/c*60*400
=3m/c
circumference=3m

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Find the value of Sin(-pi/6)
On a unit circle, starting from zero, rotate clockwise π/6 radians which gives you a reference angle of π/6 in quadrant IV where sin<0.
Sin(-pi/6)=-1/2