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when logx=2logA-(1/2)logB, x is equal to
.
logx=2logA-(1/2)logB
logx = logA^2 - logB^(1/2)
logx = log(A^2/B^(1/2))
x = A^2/B^(1/2)
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or,

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what is the equation of a circle with center (2,0) and radius 2 units?
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Equation of a circle:
(x-h)^2 + (y-k)^2 = r^2
where
(h,k) is the center
r is the radius
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Plug in given data into:
(x-h)^2 + (y-k)^2 = r^2
(x-2)^2 + (y-0)^2 = 2^2
(x-2)^2 + y^2 = 4 (this is what they're looking for)

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1 hose can fill a goldfish pond in 65 mins, and 2 hoses can the same pond in 40 mins. find how long it takes the second hose alone to fill the goldfish pond
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Let x = time (minutes) it takes for second hose alone
then
40(1/65 + 1/x) = 1
multiply both sides by 65x:
40(x + 65) = 65x
40x + 2600 = 65x
2600 = 25x
104 minutes = x

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divide 72 by 4 to get:
18 km/hr

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The sum of the measures of the angles in any triangle is 180°. Jacqueline has a triangular area in her back yard where she intends to create a garden. If the largest angle is five times the smallest and the middle angle is 20° more than twice the smallest, find the measure of each angle. (It may help to draw the triangle and label each angle.)
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Let x = measure of smallest angle
then
5x = largest angle
2x+20 = middle angle
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x + 5x + 2x+20 = 180
8x+20 = 180
8x = 160
x = 20 degrees (smallest angle)
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largest angle:
5x = 5(20) = 100 degrees
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middle angle:
2x+20 = 2(20)+20 = 40+20 = 60 degrees

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A ball is thrown upward from the top of a building. The ball's height above the ground after T seconds is given by the function: h(t) = -16t^2+48t+32.
A. What is the initial height (i.e. the height of the building)?
initial height is when t=0:
h(t) = -16t^2+48t+32
h(0) = -16(0)^2+48(0)+32
h(0) = 32 feet
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B. How high did the ball go?
vertex is at max:
time, at vertex:
t = -b/(2a)
t = -48/(2(-16))
t = -48/(-32)
t = 3/2
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Height at t=3/2:
h(3/2) = -16(3/2)^2+48(3/2)+32
h(3/2) = -16(9/4)+24(3)+32
h(3/2) = -4(9)+24(3)+32
h(3/2) = -36+72+32
h(3/2) = 68 feet
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C. When does the ball hit the ground?
set h(t) to zero and solve for t:
h(t) = -16t^2+48t+32
0 = -16t^2+48t+32
0 = t^2-3t-2
solve by applying the "quadratic formula" to get:
t = {3.56, -0.56}
throw out the negative solution (extraneous) leaving
t = 3.56 seconds
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Details of quadratic formula follows:

Solved by pluggable solver: SOLVE quadratic equation with variable

Quadratic equation (in our case ) has the following solutons: For these solutions to exist, the discriminant should not be a negative number. First, we need to compute the discriminant : . Discriminant d=17 is greater than zero. That means that there are two solutions: . Quadratic expression can be factored: Again, the answer is: 3.56155281280883, -0.56155281280883. Here's your graph:

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A lot is in the shape of a right triangle. The shorter leg measures 150m. The hypotenuse is 50m longer than the length of the longer leg. How long is the longer leg?
Let x = length (m) of longer leg
then
x+50 = length of hypotenuse
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applying Pythagorean theorem:
150^2 + x^2 = (x+50)^2
150^2 + x^2 = (x+50)(x+50)
22500 + x^2 = x^2+100x+2500
22500 = 100x+2500
20000 = 100x
200 m = x (longer leg)

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(-9x3squared+3x2squared-15x)divided by (-3x)
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we can "distribute" thus:
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L=NE/RT+NR
factoring out an 'N' on the right side:
L=N(E/RT+R)
multiplying both sides by (RT+R):
L(RT+R) = NE
LR(T+1) = NE
dividing both sides by E:
LR(T+1)/E = N

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How many quarts of pure antifreeze must be added to 2 quarts of a 10% antifreeze solution to obtain a 40% antifreeze solution? and round to the nearest tenth as needed. My email is golittlediva@yahoo.com
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Let x = amount (quarts) of pure antifreeze added
then
x + .10(2) = .40(x+2)
x + .20 = .40x+.80
x = .40x+.60
.60x = .60
x = 1.0 quart

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The diagonal of a rectangle is 1m less than twice the length of one of the sides. The other side is 5m long. What is the diagonal
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Let d = diagonal
then
d/2 = length of one side
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applying Pythagorean theorem:
d^2 = (d/2)^2 + 5^2
d^2 = d^2/4 + 25
4d^2 = d^2 + 100
3d^2 = 100
d^2 = 100/3
d = sqrt(100/3)
d = sqrt(100)/sqrt(3)
d = 10/sqrt(3)
d = 10/sqrt(3) * sqrt(3)/sqrt(3)
d = 10sqrt(3)/3 meters

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Gina has to spend 3/10 of her weekly income on food and board and 1/5 on travel to and from university and 1/8 on text books. What fraction of her income does she spend?
3/10 + 1/5 + 1/8
12/40 + 8/40 + 5/40
(12+8+5)/40
25/40
5/8 (answer)
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If Gina can save one third of her remaining income what fraction of her income does she save?
Amount remaining: 1-5/8 = 3/8
.
(1/3)(3/8)
3/24
1/8 (answer)

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One music company charges a $15 membership fee and $0.75 each to download a song. Another company charges $.90 per song to download a song (no membership fee).
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Let x = number of songs downloaded
then
1st company:
.75x + 15
2nd company:
.90x
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For what number of songs will both companies charge the same amount?
set the two equations equal to each other and solve for x:
.75x + 15 = .90x
15 = .15x
15/.15 = x
100 songs = x
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How much will it cost when each company charges the same amount?
plug above into:
.90x
.90(100)
$90

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y^3-11y^2+28y
y(y^2-11y+28)
y(y-4)(y-7)

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Two mortgage companies want to hire you as a consultant. Firm A offers you $60 plus rate of $50 per hour. Firm B offers you a flat rate of $65 per hour.
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Let x = hours worked
then
Firm A:
60x+50
Firm B:
65x
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A) Find the "break-even" point - the point where both companies would pay the same amount.
"break-even" is when they amount earned is the same for either Firm:
60x+50 = 65x
50 = 5x
10 hours = x
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B) If you work for 10 hours, which firm would give you the most money?
From part A, working 10 hours is the "break-even" point.
This is when both firms will pay exactly the same amount.
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You could plug it in and verify:
Firm A:
60x+50 = 60(10)+50 = 600+50 = $650
Firm B:
65x
65x = 65(10) = $650
Answer: both firms will give exactly the same amount

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I need to express my answer in the form P(x)=ax^+bx+c. The vertex is (-1,-4); through (5,104).
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"Vertex form" of quadratic:
y = a(x-h)^2 + k
where
(h,k) is the vertex
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since the problem gives you the vertex (-1,-4) and one point (5,104) plug it in to the vertex form and solve for 'a':
y = a(x-h)^2 + k
104 = a(5-(-1))^2 + (-4)
104 = a(5+1)^2 - 4
104 = a(6)^2 - 4
104 = a(36) - 4
108 = a(36)
3 = a
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Now, our vertex form is:
y = a(x-h)^2 + k
y = 3(x-(-1))^2 + (-4)
y = 3(x+1)^2 - 4
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we now convert the above to the correct form:
y = 3(x+1)^2 - 4
y = 3(x+1)(x+1) - 4
y = 3(x^2+2x+1) - 4
y = 3x^2+6x+3 - 4
y = 3x^2+6x-1
so,
P(x)=3x^2+6x-1 (answer)

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It takes Bob 4 hours longer to repair a car than it takes Ken. Working together, they can complete the job in 1.5 hours. How long would each of them take working alone.
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Let x = time (hours) it takes ken to do job alone
then
x+4 = time it takes bob
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1.5(1/(x+4) + 1/x) = 1
multiplying both sides by x(x+4):
1.5(x + x+4) = x(x+4)
1.5(2x+4) = x^2+4x
3x+6 = x^2+4x
6 = x^2+x
0 = x^2+x-6
0 = (x+3)(x-2)
x = {-3, 2}
throw out negative solution (extraneous) leaving:
x = 2 hours (Ken)
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Bob:
x+4 = 2+4 = 6 hours

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1)find the break-even point for the firm whose cost function C and revenue function R are given. round the intermediate calculation to nearest integer.
C(x)=210x+20,000 ; R(x)=240x
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Breakeven is when
C(x) = R(x)
210x+20000 = 240x
20000 = 30x
666.66 = x
rounded to nearest integer:
667 = x

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Sasha is planning to put a patio around her new pool which has dimensions of 20m by 30m. she wants the patio to have a uniform width,x,and to surround the entire pool. Her budget is $6000 and she knows it will cost $10/m^2 to construct the patio. How wide can she afford for the patio to be?
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Let x = width of patio
then
area of "pool and patio" is:
(2x+20)(2x+30)
4x^2+60x+40x+600
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our equation:
10((2x+20)(2x+30)- (20)(30))= 6000
10(4x^2+60x+40x+600 - 600)= 6000
10(4x^2+100x)= 6000
(4x^2+100x)= 600
4x^2+100x-600 = 0
x^2+25x-150 = 0
x^2+25x-150 = 0
(x+30)(x-5) = 0
x = {-30, 5}
throw out the negative solution (extraneous) leaving
x = 5 m

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James left home and travelled towards the moumtains.Lisa left 2.1 hours later travelling 35 km/h faster in an effort to catch him. After 1.2 hours Lisa finally caught up with him.What was James average speed?
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Let x = james average speed (km/h)
then
x+35 = lisa's average speed (km/h)
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1.2 hours = Lisa's driving time
1.2+2.1 = 3.3 hours = James' driving time
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our equation
1.2(x+35) = 3.3x
1.2x+42 = 3.3x
42 = 2.1x
42/2.1 = x
20 mph = x

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Working together, it takes two computers 15 minutes to send out a company’s email. If it takes the slower computer 60 minutes to do the job on its own, how long will it take the faster computer to do the job on its own?
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Let x = time (minutes) it takes faster computer to do job alone
then
15(1/x + 1/60) = 1
multiplying both sides by 60x:
15(60 + x) = 60x
900 + 15x = 60x
900 = 45x
20 minutes = x

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log6(x-1)+log6(x+4)=2
log6(x-1)(x+4)=2
(x-1)(x+4)=6^2
x^2+3x-4=36
x^2+3x-40=0
(x+8)(x-5) = 0
x = {-8,5}
throw out the -8 (extraneous) leaving:
x = 5

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how many liters of a 30% saline solution must be mixed with a 40% saline solution to obtain 25 liters of a 32% saline solution?
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Let x = amount (liters) of 30%
then
25-x = amount (liters) of 40%
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.30x + .40(25-x) = .32(25)
.30x + 10-.40x = 8
10-.10x = 8
-.10x = -2
x = 20 liters (of 30%)
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amount of 40%:
25-x = 25-20 = 5 liters

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A rocket shot into the aire has a height of s= 128t - 16t^2 feet, where t is the number of seconds after the rocket is shot. During what time after the rocket is shot is it atleast 240 feet high??
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set s to 240 and solve for t:
s= 128t - 16t^2
240 = 128t - 16t^2
15 = 8t - t^2
t^2 + 15 = 8t
t^2 - 8t + 15 = 0
(t-3)(t-5) = 0
t = {3, 5}
answer: the entire time between 3 seconds and 5 seconds

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Hugo has $20 . He spends n dollars. How much does he have left ?
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He has 20-n dollars left.

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craig and jim,working together, can clean the house in 2 hours. Working alone, jim takes twice as long as craig. How long does it take craig to clean the house alone?
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Let x = time (hours) it takes Craig to clean the house alone
then
2x = time (hours) it takes Jim to clean the house alone
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The equation:
2(1/x + 1/(2x)) = 1
multiplying both sides by 2x:
2(2 + 1) = 2x
2(3) = 2x
6 = 2x
3 hours = x (Time it takes Craig)

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answer the question below about the quadratic function g(x) =2x^2+8x+12
1) does the function have a minium or maxium value
Because the leading coefficient is POSITIVE, the parabola opens downwards -- thus the function has a maximum.
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It is easier to answer part 3) first:
3) where does the minimum or maxium occur
this is the "axis of symmetry":
x = -b/(2a)
x = -8/(2*2)
x = -8/(4)
x = -2
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2) what is the funtion mininum or maxium value
plug -2 into the the equation to find the max:
g(x) =2x^2+8x+12
g(-2) =2(-2)^2+8(-2)+12
g(-2) =2(4)+(-16)+12
g(-2) =8-16+12
g(-2) =-8+12
g(-2) = 4

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write the quadratic equation whose roots are -4 and -5 and whose leading coefficient is 5 use the letter x to represent the variable
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If roots are -4 and -5, then the factors are
(x+4) and (x+5)
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if you want the leading coefficient also, we multiply it altogether to get the "quadratic equation":
5(x+4)(x+5)
5(x^2+5x+4x+20)
5(x^2+9x+20)
5x^2+45x+100

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how can i get the consucative even odd digit number
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for "even consecutive" numbers:
Let x = first even number
then
x+2 = second even consecutive number
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for "odd consecutive" numbers:
Let x = first odd number
then
x+2 = second odd consecutive number
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Apply the above to your problems. It will work out...

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answer would be "B"
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take the log of both sides:
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from "logarithm rules":