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orca answered: 409 problems
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Rational-functions/200673: 3x^3-12x=
1 solutions

Answer 150952 by orca(409) on 2009-06-20 22:53:10 (Show Source):

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3x%5E3-12x

factor 3x out as it is the common factor of both terms
= 3x%28x%5E2-2%5E2%29

x^2-2^2 can be factored further
= 3x%28x%2B2%29%28x-2%29

logarithm/200367: If a^(1/3)+b^(1/3)+c^(1/3)=0, then show that log[(a+b+c)/3]=1/3[loga+logb+logc]
1 solutions

Answer 150659 by orca(409) on 2009-06-17 01:40:06 (Show Source):

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a%5E%281%2F3%29%2Bb%5E%281%2F3%29%2Bc%5E%281%2F3%29=0

a%5E%281%2F3%29%2Bb%5E%281%2F3%29=-c%5E%281%2F3%29

%28a%5E%281%2F3%29%2Bb%5E%281%2F3%29%29%5E3=%28-c%5E%281%2F3%29%29%5E3

cube both sides
a%2B3a%5E%282%2F3%29b%5E%281%2F3%29%2B3a%5E%281%2F3%29b%5E%282%2F3%29%2Bb=-c

a%2Bb%2B3a%5E%281%2F3%29b%5E%281%2F3%29%28a%5E%281%2F3%29%2Bb%5E%281%2F3%29%29=-c

a%2Bb%2Bc%2B3a%5E%281%2F3%29b%5E%281%2F3%29%28a%5E%281%2F3%29%2Bb%5E%281%2F3%29%29=0

Noting that a%5E%281%2F3%29%2Bb%5E%281%2F3%29=-c%5E%281%2F3%29

, we have

a%2Bb%2Bc%2B3a%5E%281%2F3%29b%5E%281%2F3%29%28-c%5E%281%2F3%29%29=0

a%2Bb%2Bc=3a%5E%281%2F3%29b%5E%281%2F3%29c%5E%281%2F3%29

%28a%2Bb%2Bc%29%2F3=a%5E%281%2F3%29b%5E%281%2F3%29c%5E%281%2F3%29

So
log[(a+b+c)/3]=1/3[loga+logb+logc]

Polynomials-and-rational-expressions/200269: Factor the trinomial: 8x^2-16-28x
1 solutions

Answer 150601 by orca(409) on 2009-06-16 16:58:31 (Show Source):

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8x%5E2-16-28x

re-arrange the terms in descending order
= 4%282x%5E2-7x-4%29

factor out the common factor 4
= 4%282x%2B1%29%28x-4%29

Linear-systems/200112: x^2+y^2+2x-4y-4=0 complete square and state the center and the radius or the circle
1 solutions

Answer 150434 by orca(409) on 2009-06-15 05:23:45 (Show Source):

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First moving the constant term to the right side, we have
x%5E2%2By%5E2%2B2x-4y=4

Re-arranging the terms on the left side, we get
x%5E2%2B2x%2By%5E2-4y=4

Add 1 and 4 to both sides, we have
x%5E2%2B2x%2B1%2By%5E2-4y%2B4=4%2B1%2B4

So the center of the circle is (-1, 2). The radius is 3.

Surface-area/200077: if a square has the side length 5 square root of 2 what is the area?
1 solutions

Answer 150426 by orca(409) on 2009-06-14 23:14:54 (Show Source):

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The length of the side is
a=5sqrt%282%29

It area A is
A=a%5E2=%285sqrt%282%29%29%5E2=5%5E2%2A%28sqrt%282%29%29%5E2=25%2A2=50

Inequalities/200080: What would be the correct solution to the inequality equation,:
m/4+6<1/2
1 solutions

Answer 150425 by orca(409) on 2009-06-14 23:11:32 (Show Source):

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m%2F4%2B6%3C1%2F2

Subtract 6 from both sides of the inequality
m%2F4%3C-11%2F2

multiply both sides by 4
m%3C-22

Numbers_Word_Problems/200088: Three times the opposite of a number is 40 greater than 5 times number. What is the number?
1 solutions

Answer 150423 by orca(409) on 2009-06-14 23:06:47 (Show Source):

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Suppose the unknown number is x.
Then the opposite of the number is -x.
****************************************************
Three time the opposite of the unknown number is -3x.
Five times the unknown number is 5x.
****************************************************
As their difference is 40, we have
-3x - 5x = 40
****************************************************
Solving for x, we have
-8x = 40
x = -5
****************************************************
So the number is -5.

Bodies-in-space/200090: If the height of a right circular cylinder is quadrupled and the radius is divided by three, how is the volume changed?
1 solutions

Answer 150422 by orca(409) on 2009-06-14 23:01:00 (Show Source):

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Suppose the height and the radius of the original cylinder are h and r respectively. Then the volume of the original cylinder is
v+=+pi%2Ar%5E2%2Ah

The height and the radius of the newly formed cylinder are 3h and r/3 respectively. Then the volume of the newly formed cylinder is
V+=+pi%2A%28r%2F3%29%5E2%2A3h

So the volume of the newly formed cylinder is only one third of the volume of the original cylinder.

logarithm/197349: Simplify.
log (subscript 6) 216^-3
1 solutions

Answer 147975 by orca(409) on 2009-05-22 15:11:50 (Show Source):

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Note that 216=6%5E3

216%5E-3=%28216%29%5E-3=%286%5E3%29%5E-3=6%5E-9

Therefore:
log(6)216^-3
=log(6)6^-9
=-9

Reduction-of-unit-multipliers/195943: Sue walked to her school at 2 mph, picked up her bike and rode home at 10 mph. If her round trip took 1.5 hours, how far is the school form Sue's home

1 solutions

Answer 146938 by orca(409) on 2009-05-12 02:40:11 (Show Source):

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Let x be the distance between her home and the school.
The time spent on the way to school is x/2
The time spent on the way from school to home is x/10.
As the total time is 1.5, we have
x%2F2+%2Bx%2F10=1.5

Solving the equation for x, we have
10%28x%2F2+%2Bx%2F10%29=1.5%2A10

multiply both sides by 10 to clear fractions
5x%2Bx=15

Graphs/195239: What is the y-intercept of 4x+8y=12?
1 solutions

Answer 146434 by orca(409) on 2009-05-07 00:21:15 (Show Source):

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To find the y-intercept, let x = 0,then solve the equation for y.
Letting x = 0, we have
8y = 12
Solving for y, we have
y = 12/8 = 3/2
+graph%28+300%2C+200%2C+-6%2C+5%2C+-10%2C+10%2C+%2812-4x%29%2F8%29+

+graph%28+300%2C+200%2C+-6%2C+5%2C+-10%2C+10%2C+%2812-4x%29%2F8%29+

Quadratic_Equations/194506: Solve using quadratic methods.
x^2/3-3x^1/3-10=0
1 solutions

Answer 145970 by orca(409) on 2009-05-03 22:55:47 (Show Source):

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Let x%5E%281%2F3%29+=+y

then the equation becomes
y%5E2-3y-10=0

Solving for y, we have
(y-5)(y+2)=0
y = 5 or y = -2
From x%5E%281%2F3%29+=5

, we get x=125
From x%5E%281%2F3%29+=-2

, we get x = -8

Radicals/194529: simplify
4/the square root of 10
1 solutions

Answer 145968 by orca(409) on 2009-05-03 22:50:29 (Show Source):

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4%2Fsqrt%2810%29

4sqrt%2810%29%2F%28sqrt%2810%29%2Asqrt%2810%29%29

Square-cubic-other-roots/194357: please help Multiplying radicals and simplify
1) (3 square2)(-4 square 10)
2)(3-2 square 7)(3 +2 square 7)
1 solutions

Answer 145858 by orca(409) on 2009-05-02 22:29:32 (Show Source):

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1)
%283sqrt%282%29%29%28-4sqrt%2810%29%29

2)
Use the formula %28a%2Bb%29%28a-b%29=a%5E2-b%5E2

to simplify
%283-2sqrt%287%29%29%283+%2B2sqrt%287%29%29

Quadratic_Equations/194368: Factoring for quadratic equations. 6x^ + 3x - 18=0

1 solutions

Answer 145857 by orca(409) on 2009-05-02 22:21:57 (Show Source):

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Note that 3 is a common factor of the three terms, we factor it out.
3%282x%5E2%2Bx-6%29=0

Next factor 2x%5E2%2Bx-6

So 2x - 3 = 0 or x + 2 = 0
From 2x-3 = 0 we get x = 3/2
From x+2=0 we get x = -2

Radicals/193954: SIMPLIFY
1/1 + THE SQUARE ROOT OF 3=
1 solutions

Answer 145605 by orca(409) on 2009-04-30 16:06:37 (Show Source):

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1%2F%281%2Bsqrt%283%29%29

%281-sqrt%283%29%29%2F%28%281%2Bsqrt%283%29%29%281-sqrt%283%29%29%29

%281-sqrt%283%29%29%2F%281%5E2-%28sqrt%283%29%29%5E2%29

Sequences-and-series/193992: Which is the next number in the series:
-1,0,3,8,15,24,35,...
1 solutions

Answer 145604 by orca(409) on 2009-04-30 16:00:43 (Show Source):

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Note that:
0 -(-1) =1
3 - 0 = 3
8 - 3 = 5
15-8 = 7
24-15 =9
35-24=11
.......
the differences are consecutive odd numbers.
Therefore the next number - 35 will be 13.
so the number is 48

Radicals/193998: solve the following equation
(√2 +√3)(√2-√3)=
1 solutions

Answer 145603 by orca(409) on 2009-04-30 15:54:01 (Show Source):

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Use the formula %28a%2Bb%29%28a-b%29=a%5E2-b%5E2

%28sqrt%282%29%2Bsqrt%283%29%29%28sqrt%282%29-sqrt%283%29%29

%28sqrt%282%29%29%5E2-%28sqrt%283%29%29%5E2

Exponential-and-logarithmic-functions/193849: rewite 3(ln)4 as the logarithm of a single number
1 solutions

Answer 145543 by orca(409) on 2009-04-29 22:34:14 (Show Source):

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Use the property of logarithm
alnb=lnb%5Ea

Expressions-with-variables/193670: The sum of two numbers is 50. If twice the larger is subtracted by 4 times the smaller, the result is 8. Find the numbers.
1 solutions

Answer 145367 by orca(409) on 2009-04-29 02:50:38 (Show Source):

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Let the smaller one be x, then the larger one is 50 - x.
Now use the given information" twice the larger is subtracted by 4 times the smaller, the result is 8 " to set up an equation:
2(50-x)-4x=8
Solving for x, we have
100 - 2x - 4x = 8
100-6x=8
100-8=6x
92=6x
6x=92
x=92/6=46/3
So the smaller number is 46/3
The larger one is 50-46/3=104/3

Quadratic_Equations/193566: I'm stuck on this one:
determine whether the following equation has a real or complex solution
√2x^2 -4x-7√2=0
1 solutions

Answer 145359 by orca(409) on 2009-04-28 23:25:34 (Show Source):

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You need to find the value of the discriminant of the equation.
If the discriminant > = 0, then the equation has real solutions. Otherwise only complex solutions.
Recall that
The discriminant = b%5E2+-+4ac

for equation ax%5E2%2Bbx+%2Bc=0

So
The discriminant =
= %28-4%29%5E2-4%2Asqrt%282%29%2A%28-7sqrt%282%29%29

> 0
So the equation has 2 real roots

expressions/193581: 5^-1 + 2^-1-3^0.
ANSWER: -3/10
anyone knows why the answer is -3/10 please give a full break down
1 solutions

Answer 145358 by orca(409) on 2009-04-28 23:18:40 (Show Source):

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Note that:
a%5E%28-1%29=1%2Fa

And

real-numbers/193449: This question is from textbook
is the quotient of two irrational numbers always an irrational number? why or why not?
1 solutions

Answer 145229 by orca(409) on 2009-04-28 01:10:44 (Show Source):

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consider two irrational numbers 2sqrt%283%29

and

Their quotient %282sqrt%283%29%29%2Fsqrt%283%29

is a rational number.

Age_Word_Problems/193438: Tom buys $20,000 worth of stock, and sells it eight months later for $28,000. If he is required to pay 15% capital gains tax on his profits, how much tax will he pay?
1 solutions

Answer 145224 by orca(409) on 2009-04-27 22:50:01 (Show Source):

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Note that the capital gain tax = profit*rate
The profit = 28,000 - 20,000 = 8,000
The tax to be paid = 8,000*15% = 1,200

Rectangles/193291: The length of a rectangle swimming pool is 2x-1 meters and the width is x+2 meters. Write a polynomial A(x) that represent the area. Find A(5)
1 solutions

Answer 145110 by orca(409) on 2009-04-27 04:18:57 (Show Source):

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Use the formula:
Area of a rectangle = length*width
So
A = %282x-1%29%28x%2B2%29

A can also be written in expanded form
A = 2x%5E2-x%2B4x-2

Polynomials-and-rational-expressions/193287: This question is from textbook
factor the trinomial
x2+7x+12
1 solutions

Answer 145109 by orca(409) on 2009-04-27 04:14:01 (Show Source):

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Note that 12 can be factored into 3*4. Because the sum of the two factors is 7 which is the coefficient of the second term, the quadratic can be factored into:
x2%2B7x%2B12

Triangles/193233: Hi,
Thank you for the help.
Layout for a soccer field: side A is 36' side B is 84'. A & B meet at 90o angle. what is the length of side C?
Also like to know the formula for determining the length. thanks! This is a real life scenario.
Gary B.
1 solutions

Answer 145089 by orca(409) on 2009-04-26 23:35:38 (Show Source):

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We are required to solve a right triangle. So apply the Pythagorean Theorem to it.
The theorem states that the sum of the squares of the two legs a aand b is equal to the square of the hypotenuse c. Or a%5E2+%2B+b%5E2+=+c%5E2

Here we are given
One leg = 36
Another leg = 84
We are required to find the length of the hypotenuse
According to the Theorem,
c%5E2+=+36%5E2+%2B+84%5E2

So
c =

= 91

Geometry_Word_Problems/193102: A rectangular picture frame is 4 times as wide as its length with a perimeter of 80 inches. Find the dimensions.
1 solutions

Answer 144977 by orca(409) on 2009-04-26 01:42:19 (Show Source):

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Let x be the length.
Then its width 4x
So the perimeter in terms of x is 2(x + 4x)
Setting it equal to 80, we have
2(x + 4x) = 80
Solving for x, we have
2*5x = 80
10x = 80
x = 8
Therefore the length is 8
the width is 4x = 4*8 = 32

Radicals/192879: V7 TIMES V35
1 solutions

Answer 144764 by orca(409) on 2009-04-24 00:45:19 (Show Source):

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sqrt%287%29%2Asqrt%2835%29

Average/192702: Factor each polynomial completely. Please show work.
a^2b + 2ab - 15b
1 solutions

Answer 144638 by orca(409) on 2009-04-23 03:11:01 (Show Source):

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Note that b is the common factor of the three terms, we must factor it out first.
a%5E2b+%2B+2ab+-+15b

Next factor a^2+2a-15.
a%5E2b+%2B+2ab+-+15b

test/192300: Two airplanes left the same airport traveling in opposite directions. If one airplane averages 400 miles per hour and the other airplane averages 250 miles per hour,in how many hours will the distance between the two planes be 1625 miles?
1 solutions

Answer 144347 by orca(409) on 2009-04-21 02:24:37 (Show Source):

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Suppose that in x hours the distance between the two planes will be 1625 miles.
In x hours, the distance traveled by each plane is:
Plane 1: 400x
Plane 2: 250x
The total distance covered is 400x + 250x.
Setting it equal to 1625, we have
400x + 250x = 1625
Solving for x, we have
650x = 1625
x= 2.5 hours

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