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L/W = 8/5 is the golden ratio.

The length of a painting is 2 feet more than the width.

This makes L = W + 2

Golden ratio is L/W = 8/5

If L = W + 2, then the golden ratio becomes (W+2)/W = 8/5

Multiply both sides of this equation by 5 to get:

5 * (W + 2)/W = 8

Multiply both sides of this equation by W to get:

5 * (W + 2) = 8*W

Remove parentheses to get:

5W + 10 = 8W

Subtract 5W from both sides of this equation to get:

3W = 10

Divide both sides of this equation by 3 to get:

W = 10/3

If W = 10/3, then L = W + 2 = 10/3 + 2 = 10/3 + 6/3 = 16/3

L = 16/3
W = 10/3

L/W = (16/3)/(10/3) = (16/3) * (3/10) = 16/10 = 8/5

You have the golden ratio when:

L = 16/3
W = 10/3

L = W + 2 means L = 10/3 + 2 means L = 10/3 + 6/3 means L = 16/3

Both requirements are met.

L = W + 2
L/W = 8/5

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you take a ratio of the area of the icon compared to the area of the computer and then apply it to the wall chart.

The wall chart is 6 feet by 6 feet.

Conver that to inches and the wall chart is 72 inches by 72 inches = 5184 square inches.

each icon on the computer is 3/8 * 3/8 = .140625 square inches.

Not sure what size computer you are using, but my computer screen measures approximately 13 inches across and 8 inches up.

That make the area of my screen 8 * 13 = 104 square inches.

The ratio of the area of an icon of .140625 square inches to the area of a typical computer screen (assuming mine is typical - yours might be different) would be .140625 / 104 = .001352163.

Apply that ratio to a screen size of 5184 square inches and you get .001352163 * 5184 = 7.009615385.

Take the square root of that to get 2.647567824 inches by 2.647567824 inches.

Round that off and your icons should be about 2.65 inches by 2.65 inches.

Let's see how that would work out.

The wall chart would hold approximately 5184 / (2.65)^2 icons which would make about 752 icons.

My computer would hold about 104 / (3/8)^2 icons which would make about 739 icons.

That's pretty close.

I would say make your icons about 2.65 inches by 2.65 inches and you would be in the ball park.

Anything between 2.5 and 3 inches by 2.5 and 3 inches should be close enough.

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144 = -12 * (x + 5)

Remove the parentheses by multiplying each term within the parentheses by -12 to get:

144 = (-12 * x) + (-12 * 5)

Simplify this by multiplying out the terms and combining like terms to get:

144 = -12*x - 60

add 60 to both sides of this equation to get:

144 + 60 = -12*x

combine like terms to get:

204 = -12*x

divide both sides of this equation by (-12) to get:

204 / (-12) = (-12) * x / (-12)

Simplify and combine like terms to get:

-17 = x

You have x = -17 as your answer.

replace this value for x in the original equation as shown below:

original equation is 144 = -12 * (x + 5)

replace x with -17 to get:

144 = -12 * (-17 + 5)

combine like terms within parentheses to get:

144 = -12 * (-12)

simplify and combine like terms to get:

144 = 144

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sale price = .7 * original price.

7% tax is added to the sale price.

total bill came to $97.54

This means that .7 * original price + .07 * .7 * original price = 97.54

let x = original price.

this formula becomes:

.7*x + .07*.7*x = $97.54

factor out the x to get:

(.7 + .07*.7) * x = $97.54

simplify and combine like terms to get:

.749 * x = $97.54

divide both sides of equation by .749 to get:

x = 97.54 / .749

simplify to get:

x = 130.2269693

original price of the radio is $130.2269693

this is not an even number, but it works.

take 30% off that to get a sale price of $91.1588785

Add 7% sales tax to that to get a total bill of $97.54

Normally these types of problems will get you an even number for the original price but this one doesn't.

Despite that, the method looks good and so I believe this must be the answer you are looking for.

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if a * b * c = 0, then either a can be equal to 0 or b can be equal to 0 or c can be equal to 0 and the equation will be true because you will get 0 = 0.

take each factor in turn and set it equal to 0 and solve for the unknown variable.

2x-3 = 0
add 3 to both sides to get:
2x = 3
divide both sides by 2 to get
x = 3/2

that's solution D.

x-11 = 0
add 11 to both sides.
x = 11

that's solution B.

x/3 + 7 = 0
subtract 7 from both sides to get:
x/3 = -7
multiply both sides by 3 to get:
x = 21

that's solution A.

Look's like the solution that is not a solution to your problem is C.

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c + 4 = -14
subtract 4 from both sides of the equation to get:
c = -18

to check your equation, replace c with -18 in your equation to see if it is true.

your original equation is c + 4 = -14

replace c with -18 to get:

-18 + 4 = -14

add 18 to both sides of this equation to get:

4 = -14 + 18

simplify and combine like terms to get:

4 = 4

since this is true (4 really does equal to 4), then your value of c = -18 is good.

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|x+y| = 1 means that:

If the expression (x+y) >= 0, then:

x + y = 1

this means that:

x = 1-y

Example:

Let y = 5
x = 1-5 = -4

|-4 + 5| = |1| = 1 which is true so this looks good.

We have so far:

x = 1-y when the expression (x+y) is greater than or equal to 0.

If the expression (x+y) < 0, then:

- (x+y) = 1

Multiply both sides of this by (-1) to get:

(x+y) = -1

this means that:

x = -1 - y

Example:

Let y = 5

This means that x = -1 - 5 = -6

If x = -6 and y = 5, we get:

(x+y) = (-6 + 5) = -1 < 0

We also get:

|x + y| = |-6 + 5| = 1 which becomes |-1| = 1 which becomes 1 = 1 so this is good.

Your solution should be, if I understand the problem correctly:

x = 1-y
or:
x = -1-y

Let's see how that holds up.

Let y be any number.

We'll try -5 and 5

When y = -5, x will be either:

1-(-5) = 6
or:
-1-(-5) = 4

If x = 6 and y = -5, then |x+y| = |6-5| = |1| = 1 which is good.

If x = 4 and y = -5, then |x+y| = |4 -5| = |-1| = 1 which is good.

When y = 5, x with be either:

1 - 5 = -4
or:
-1 - 5 = -6

If x = -5 and y = 5, then |x+y| = |-4 + 5| = |1| = 1 which is good.

If x = -6 and y = 5, then |x+y| = |-6 + 5| = |-1| = 1 which is good.

Your answer is:

x = 1-y
or:
x = -1-y

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They are definitely not the same thing.

Your answer is incorrect.

Only thing we need to do is find out why.

The equation was h(x) = 2x^3 + 3.

We will let g(x) represent the inverse function of h(x).

In order to find the inverse function, solve for x and then replace x with g(x) and replace h(x) with x.

Let's do that and see how it works out.

Original equation is h(x) = 2x^3 + 3

Subtract 3 from both sides of the equation to get:

h(x) - 3 = 2x^3

Divide both sides of the equation by 2 to get:

(h(x) - 3)/2 = x^3

take the cube root of both sides of the equation to get:

= x

Replace x with g(x) and replace h(x) with x to get:

= g(x)

g(x) is the inverse function of h(x) and agrees with your book.

You can make g(x) = and your equation becomes:



I'm not sure how you got the answer you got.

Try it again and see if it make more sense.

If you have any questions, send me an email.

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Sum of exterior angles is equal to 360 degrees always.

Sum of interior angles is equal to Number of Side * (180) * (n-2).

Example:

Sum of interior angles of a triangle = 180 * (3-2) = 180 degrees.

Sum of interior angles of a rectangle = 180 * (4-2) = 360 degrees.

Sum of interior angles of a pentagon = 180 * (5-2) = 540 degrees.

Your question was:

How many sides has a regular polygon whose interior angle is 11 times it's exterior angle.

Let x = the interior angle

Let y = the exterior angle.

Let n = number of sides of the triangle.

Each interior angle of a regular polygon = (n-2) * 180 / n

Example:

Each interior angle of a regular triangle = 1 * 180 / 3 = 60 degrees.

Each interior angle of a regular rectangle = 2 * 180 / 4 = 360 / 4 = 90 degrees.

Each exterior angle of a polygon = 360 / n

Example:

Each exterior angle of a triangle = 360 / 3 = 120 degrees.

Each exterior angle of a rectangle = 360 / 4 = 90 degrees.

The sum of the interior angle and it's exterior angle always equals 180 degrees.

For the triangle, interior angle of 60 degrees + exterior angle of 120 degrees = 180 degrees.

For the rectangle, interior angle of 90 degrees + exterior angle of 90 degrees = 180 degrees.

For the pentagon, each interior angle = (5-3) * 180 / 5 = 108 degrees.

Each exteriof angle of the pentagon = 360 / 5 = 72 degrees.

Sum of interior angle of 108 degrees and exterior angle of 72 degrees = 180 degrees.

Your problem states that the interior angle is 11 times its exterior angle.

The interior angle is equal to (n-2) * 180 / n

The exterior angle is equal to 360/n

Interior angle = 11 * exterior angle means that:

(n-2) * 180 / n = 11 * (360/n)

which says that each interior angle is equal to 11 times each exterior angle.

Solve for n:

Remove Parentheses to get:

(180 * n - 360)/n = 11 * (360/n)

Multiply both sides of equation by n to get:

180*n - 360 = (11*360)/n * n

Simplify to get:

180*n - 360 = 11 * 360

Add 360 to both sides to get:

180*n = 11*360 + 360

Simplify to get:

180*n = 12*360 = 3600 + 720 = 4320

Divide both sides by 180 to get:

n = 4320/180 = 432/18 = 24.

Number of sides of the polygon = 24.

Each interior angle = (24-2) * 180 / 24 = 22 * 180 / 24 = 165 degrees.

Each exterior angle = 180 - 165 = 15.

165/15 = 11 so this part is good (interior = 11 * exterior)

15 * 24 = 360 so sum of exterior angles is 360 which is good.

165 * 24 = 3960

Sum of interior angles of a regular polygon = (n-2) * 180.

For a polygon with 24 sides, this becomes (24-2)*180 = 22 * 180 = 3960 so this part is also good.

Your answer is that the polygon has 24 sides.

Each interior angle is 165 degrees.

Each exterior angle is 15 degrees.

Each interior angle is 165/15 = 11 times each exterior angle.

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500 - 20% of 500 = 500 - .2*500 = 500 = 100 = 400

x% of anythikng is equal to x%/100% of that thing.

Example:
10% of x = .1 * x

20% of y = .2 * y

etc.

I would need a specific problem to go any further because I'm not sure what you are asking.

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x = retail price
.2*x = discount off retail price

.2x = 45
divide both sides of equation by .2
x = 45/.2 = 225.00

Retail Price was $225.00
.2 * $225.00 = $45.00 = discount.
Discount Price = Retail Price minus Discount = $225.00 - $45.00 = $180.00

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Easiest way is to convert both equations to slope-intercept form of y = mx + b where m is the slope and b is the y-intercept.

First equation is:

y = -(1/2)x - 11

It is already in slope intercept form.

Second equation is

16x - 8y = -8
subtract 16x from both sides of the equation to get:
-8y = -16x - 8
divide both sides of the eqution by -8 to get:
y = 2x + 1

If the slopes are equal they would be either identical or parallel to each other.
If the slopes are negative reciprocals of each other, they would be perpendicular to each other.

They are not identical or parallel to each other.

Take negative reciprocal of (-1/2) to get 2.

Slopes are negative reciprocals of each other so these line should be identical.

Prove by graphing.

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I believe the formula you are looking to solve is:



If so, then the solution would appear to be as follows:

multiply both sides of the equation by x^2 to get:

1 <= x^2

This is the same as x^2 >= 1

Take the square root of both sides of this equation to get:

x >= 1
or:
x <= -1

How I got to this part I can't really explain that well, but it works.

square root of x^2 is x.

No problem there, because x * x = x^2 regardless if x is positive or negative.

If it was = rather than >=, the answer would have been:

x = +/- 1

Because it was >=, then you have to take into account that when you multiply both sides of an inequality by -1, then the inequality reverses.

This causes:

x >= 1
or
x <= -1

To confirm the answer is good, you would take some values of x and replace x in the original equation with them.
Take values in and out of the acceptable range.

Let x equal:
-2,-1,-.5,0,.5,1,2

Then 1/x^2 <= 1 becomes:

When x = -2, 1/x^2 = 1/4 < 1 = ok.
When x = -1, 1/x^2 = 1/1 = 1 = ok
When x = -.5, 1/x^2 = 1/.25 > 1 = NOT ok.
When x = 0, 1/x^2 = 1/0 = undefined = NOT ok.
When x = .5, 1/x^2 = 1/.25 > 1 = NOT ok.
When x = 1, 1/x^2 = 1/1 = 1 = ok.
When x = 2, 1/x^2 = 1/4 < 1 = ok.

Your answer is that x >= 1 or x <= -1.

You do not have to restrict the domain to eliminate division by 0 because the domain already excludes 0.

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16 < s^2 < 60
Let s = length of one of the sides of your square.
If s = 4, then s^2 = 16
If s = 5, then s^2 = 25
If s = 6, then s^2 = 36
If s = 7, then s^2 = 49
If s = 8, then s^2 = 64

Looks like your integers would be:

4,5,6,7

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You have to know what percentage of your final grade is the project.

Assume 25%

Your formula would be:

.25 * project grade plus .75 * the average of your other grades = final grade.

Assume this percentage is correct, then your final grade would be:

.25 * 100% + .75 * 68% = 76%

Your answer will vary depending on what percentage of your final grade is assigned to the project.

If a 20 point project means it's worth 20% of your grade, then adjust these figures accordingly.

This is a question that your teacher would be in the best position to answer.

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Your polynomial is 4-5b.

If you multiply this polynomial by (-1) * (-1) it will remain the same because (-1)^2 = 1

your expression becomes:

(-1) * (-1) * (4-5b).

If you multiply (-1) * (4-5b) you get (-4+5b) which is the same as (5b-4)

your expression becomes:

(-1) * (5b-4)

You have just factored out (-1) from your polynomial expression of (4-5b)

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Your equation to solve is:

sin^4(x)-cos^4(x)/sin^2(x)=1-cot^2(x)

Multiply both sides of this equation by sin^2(x) to get:

sin^4(x)-cos^4(x) = (1-cot^2(x)) * sin^2(x)

Remove parentheses on the right side of your equation to get:

sin^4(x)-cos^4(x) = sin^2(x) - (cot^2(x)) * sin^2(x))

Since cot(x) = cos(x) / sin(x), and (cos(x)/sin(x))^2 = cos^2(x)/sin^2(x), your equation becomes:

sin^4(x)-cos^4(x) = sin^2(x) - (cos^2(x)/sin^2(x)) * sin^2(x))

Simplify the right side of your equation to get:

sin^4(x)-cos^4(x) = sin^2(x) - cos^2(x)

Since sin^4(x) - cos^4(x) = (sin^2(x) + cos^2(x)) * (sin^2(x) - cos^2(x)), your equation becomes:

(sin^2(x) + cos^2(x)) * (sin^2(x) - cos^2(x)) = sin^2(x) - (cos^2(x)

Since sin^2(x) + cos^2(x) = 1, your equation becomes:

sin^2(x) - cos^2(x) = sin^2(x) - (cos^2(x)

Since this is what you wanted to prove, you're done.

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Your expression is

Since , your expression becomes:



This becomes:



Since , this means that:



Replace in your expression with this to get:



Since , you can replace cos(x) in your equation to get:



This is the same as:

because .

Your expression has become:



Remove parentheses to get:



Since , you can replace in your equation to get:



Simplify to get:



Since this is what you wanted to prove, you're done.

Note I am showing sin^2(x) as sin(x)^2 because it doesn't come out good on the formula rendering routine with sin^2(x).

Example:

sin^2(x) shows up as

sin(x)^2 shows up as

The second version is technically incorrect but it shows up clearer so I used it.

Note that sin(x) * sin(x) really is (sin(x))^2 but I shortened it to sin(x)^2 to eliminate all those extra parentheses that muddied up the presentation.

Just remember that sin(x)^2 is the same as sin^2(x) and we'll be ok.

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3x - y = 6 (equation 1)
-4x + 2y = -8 (equation 2)

Solve by substitution.

Take equation 1 and solve for y.

3x - y = 6 (equation 1)
add y to both sides of the equation to get:
y + 6 = 3x
subtract 6 from both sides of the eqution to get:
y = 3x - 6

You have now solved for y in terms of x in equation 1.

Take equation 2 and replace y with 3x - 6. This is the substitution part.

-4x + 2y = -8 (equation 2)
Replace y with value of y calculated from equation 1.
-4x + (2 * (3x-6)) = -8
Remove parentheses.
-4x + 2*3x - 2*6 = -8
Simplify.
-4x + 6x - 12 = -8
Combine like terms.
2x - 12 = -8
Add 12 to both sides of the equation.
2x = -8 + 12
Simplify and combine like terms.
2x = 4
Divide both sides of the equation by 2.
x = 4/2 = 2

You have now solved for x in equation 2.

Take the value of x and replace x with it in equation 1 to solve for y.

3x - y = 6 (equation 1)
Replace x with 2.
3*2 - y = 6
Simplify.
6 - y = 6
Subtract 6 from both sides of the equation.
-y = 0
Multiply both sides of the equation by -1.
y = 0

You have now solved for y in equation 1.

The values that you calculated for both equations are:

x = 2
y = 0

Replace x and y in both original equations to see if these values will solve both equations at the same time (simultaneously).

3x - y = 6 (equation 1)
-4x + 2y = -8 (equation 2)

Replace x with 2 and y with 0 in both equations.

3x - y = 6 in equation 1 becomes 6 - 0 = 6 becomes 6 = 6 which is true.

-4x + 2y = -8 in equation 2 becomes -8 = 0 = -8 becomes -8 = -8 which is true.

Both equations are true indicating the values for x and y that you calculated will solve both equations simultaneously.

x = 2 and y = 0 are your answers for solving these two equations simultaneously.

You did it through the process of substitution.

Now we'll do elimination.

Same two equations are:

3x - y = 6 (equation 1)
-4x + 2y = -8 (equation 2)

Object is to manipulate each equation by multiplying or dividing as necessary (mostly multiplying) so that one of the variables will have a common coefficient in both equations. When you add or subtract one equation from the other, the variable with the common coefficients will cancel out and you will be left with one equation in one unknown variable that you can solve.

We will multiply equation 1 by 2.

6x - 2y = 12 (equation 1 multiplied by 2)
-4x + 2y = -8 (equation 2)

Because the variable y in each equation has the same coefficients and the signs are different, we'll add equation 2 to equation 1 in order to eliminate the y variable.

6x - 2y = 12 (equation 1 multiplied by 2)
plus:
-4x + 2y = -8 (equation 2)
equals:
2x = 4
Divide both sides of this equation by 2.
x = 2

We have solved for x.

This was done through the process of elimination.

Now we go back to equation 1 and replace x with 2 and solve for y.

These are the same steps you did in the substitution part. Once you get to this point, the remaining operations are the same regardless if you got here through substition or elimination.

3x - y = 6 (equation 1)
Replace x with 2.
3*2 - y = 6
simplify.
6 - y = 6
Subtract 6 from both sides of the equation and add y to both sides of the equation.
y = 0

We have solved for y in equation 1.

We have:

x = 2
y = 0

Now we confirm in all three equations and we're done.

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Let P = speed of the plane.
Let W = Speed of a full headwind.
Let P = Speed of the plane.
Let R1 = Overall speed going against a full headwind = P-W
Let R2 = Overall speed going against half a full headwind = P-(W/2)
Let T1 = Amount of time it takes against a full headwind.
Let T2 = Ampount of time it takes against half a full headwind.
Let D = distance.

Distance is 2900 miles
With a full headwind it takes 5 hours.
With half a full headwind it takes 4 hours 50 minutes = 4.8333333 hours.

We have:

T1 = 5 hours
D = 2900
T2 = 4.8333333 hours

Rate * Time = Distance

Formula against a full headwind is:

R1 * T1 = D which becomes:

R1 * 5 = 2900

Since R1 = P-W, this formula becomes:

(P-W) * 5 = 2900 (equation 1)

Formula against half a full headwind is:

R2 * T2 = D becomes:

R2 * 4.8333333 = 2900

Since R2 = (P-W/2), this formula becomes:

(P-W/2)*4.8333333 = 2900 (equation 2)

We can solve for (P-W) to get:

(P-W) = 2900 / 5 = 580 miles per hour.

We can solve for (P-W/2) to get:

(P-W/2) = 2900 / 4.8333333 = 600 miles per hour.

Since (P-W) = 580, we can solve for P to get:

P = W + 580 (equation 3)

Since (P-W/2) = 600, we can solve for P to get:

P = W/2 + 600 (equation 4)

Since equation 3 and equation 4 both equal to P, then they both equal to each other and we get:

W + 580 = W/2 + 600

Multiply both sides of this equation by 2 to get:

2W + 1160 = W + 1200

Subtract W from both sides of this equation and subtract 1160 from both sides of this equation to get:

2W - W = 1200 - 1160

Combine like terms and simplify to get:

W = 40

Substitute 40 for W in equation 3 to get:

P = W + 580 (equation 3) becomes:

P = 40 + 580 = 620

So far we have:

P = 620
W = 40

To confirm these answers are correct, we substitute in equations 1 and 2 to get:

(P-W) * 5 = 2900 (equation 1) becomes:
(620-40) * 5 = 2900 which becomes:
580 * 5 = 2900

Divide both sides of this equation by 5 to get:

580 = 2900/5 = 580 which is true so answer is confirmed for equation 1.

(P-W/2)*4.8333333 = 2900 (equation 2) becomes:
(620-20)*4.8333333 = 2900 which becomes:
600*4.8333333 = 2900

Divide both sides of this equation by 4.8333333 to get:

600 = 2900 / 4.8333333 = 600 which is true so answer is confirmed for equation 2.

Answer to the question is:

Airplane speed is 620 miles per hour.
Headwind speed is 40 miles per hour.

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S = smaller number
L = larger number

S = 2L/3 (equation 1)

1/S + 1/L = 1/6 (equation 2)

Multiply both sides of equation 2 by S*L.

1/S + 1/L = 1/6 becomes:

L + S = (S*L)/6

Multiply both sides of this equation by 6 to get:

6L + 6S = S*L

Replace S with (2L)/3 to get:

6L + 6*(2L)/3 = L*2L/3

Remove Parentheses to get:

6L + 12L/3 = 2L^2/3

Multiply both sides of equation by 3 to get:

18L + 12L = 2L^2

Combine like terms to get:

30L = 2L^2

Divide both sides of equation by 2 to get:

15L = L^2

Subttract 15L from both sides of equation to get:

L^2 - 15L = 0

Factor L to get:

L*(L-15) = 0

Solve for L to get:

L = 0
or:
L = 15

Substitute these in original equations to get:

S = 2L/3 becomes:

S = 2*0/3 = 0
or:
S = 2*15/3 = 10

When S = 0 and L = 0, then 1/S + 1/L = 1/6 becomes:

1/0 + 1/0 = 1/6 which is false so L = 0 is not a valid answer and can be rejected.

When S = 10 and L = 15, then 1/10 + 1/15 = 1/6.

Multiply both sides of of this equation by 30 to get:

3 + 2 = 5 which is true so L = 15 is good.

Your answer is L = 15 and S = 10.

The larger number is 15 and the smaller number is 10.

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I did a little research for you at the following website:

Newton's formula of cooling or heating

The formula is:

T(t) = S + (T(0) - S)*e^kt

where:

t = time in hours
T(0) = initial temperature of the turkey
T(t) = final temperature of the turkey after t hours.
S = temperature of the surrounding medium.
k = constant rate of cooling or heating.

In your problem:

t = 4 hours
T(0) = 40 degrees for the turkey
T(t) = 170 degrees of the turkey after 4 hours in the oven.
S = temperature of surrounding medium = 350 degree oven.
k = constant of cooling or heating which is what you want to find.

We plug these values into the formula as shown below and solve for k.

T(t) = S + (T(0)-S)*e^kt becomes:

170 = 350 + (40-350) * e^4k

This becomes:

170 = 350 - (310 * e^4k)

subtract 350 from both sides of the equation to getr:

170 - 350 = -310 * e^(4k)

Simplify to get:

-180 = -310*e^(4k)

Divide both sides of the equation by -310 to get:

-180/-310 = e^(4k)

Take natural log of both sides to get:

ln(-180/-310) = ln(e^(4k))

This becomes:

ln(-180/-310) = 4k * ln(e)

Since ln(e) = 1, this becomes:

ln(-180/-310) = 4k

Divide both sides of this equation by 4 to get:

ln(-180/-310) / 4 = k

Solve for k to get:

k = -.135903862

Plug this back into the equation to get:

170 = 350 + (40-350) * e^4*-.135903862

Solve this to get:

170 = 170

Now that we have a value for k, we can start all over with the formula to find out the remaining time.

The formula is:

T(t) = S + (T(0)-S)*e^kt

where:

t = time in hours
T(0) = initial temperature of the turkey
T(t) = final temperature of the turkey after t hours.
S = temperature of the surrounding medium.
k = constant rate of cooling or heating.

We have:

t = time in hours we want to find.
T(0) = initial temperature of the turkey = 170 degrees.
T(t) = final temperature of the turkey after t hours = 185 degrees.
S = temperature of the surrounding medium = 350 degrees of the oven.
k = constant rate of cooling or heating = -.135903862

Formula becomes:

185 = 350 + (170-350)*e^-.135903862*t

We solve for t to get:

-165 = -180*e^-.135903862*t

Divide both sides of this equation by -180 to get:

-165/-180 = e^-.135903862*t

Take natural log of both sides of this equation to get:

ln(-165/-180) = ln(e^-.135903862*t)

Because log(b^n) = n*log(b), this becomes:

ln(-165/-180) = -.135903862*t*ln(e)

Because ln(e) = 1, this becomes:

ln(-165/-180) = -.135903862*t

Divide both sides by -.135903862 to get:

t = ln(-165/-180)/-.135903862

solve for t to get:

t = .640242124 hours

That should be your answer.

To confirm this calculation is good, I went back to the beginning and plugged 4.640242124 hours into the original equation to get:

T(t) = S + (T(0) - S)*e^kt becomes:

185 = 350 + (40-350)*e^4.640242124*-.135903862

Solving the right side of the equation, I get:

185 = 350 + (40-350)*e^4.640242124*-.135903862 = 185.

This confirms the answer is good because we went all the way from 40 degrees to 185 degrees without stopping at 170 degrees and the total time becomes the same as if we went from 40 to 170 and then 170 to 185.

Your answer is:

t = .640242124 hours to go from 170 degrees to 185 degrees.

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Your equation to solve is:

20x^(2/3) - 6x^(1/3) - 2 = 0

If you let y = x^(1/3), then y^2 = x^(1/3)^2 = x^(2/3)

Your equation would become:

20y^2 - 6y - 2 = 0

This might be able to be factored to something like:

(5y + 1) * (4y - 2)

To test this out, do the multiplications of the factors as shown below:

-2 * 1 = -2
-2 * 5y = -10y
4y * 1 = 4y
4y * 5y = 20y
Add these together and combine like terms and you get:

20y - 10y + 4y - 2 becomes:
20y - 6y - 2

So you have found the factors of:

(5y + 1) * (4y - 2) = 0

This results in:

5y + 1 = 0 which results in y = -.2
4y - 2 = 0 which results in y = .5

Now you had made y = x^(1/3) so this equation becomes:

x^(1/3) = -.2
and:
x^(1/3) = .5

x^(2/3) = x^(1/3)^2 = -.2^ = .04
and:
x^(2/3) = x^(1/3)^2 = .5^ = .25

We hold these values to test against the original equation.

First values are x^(1/3) = -.2 and x^(2/3) = .04

Plug into the original equation of:

20x^(2/3) - 6x^(1/3) - 2 = 0 to get:

20*.04 - 6*(-.2) - 2 = 0
simplify to get:
.8 + 1.2 - 2 = 0
combine like terms to get:
0 = 0 which is true so it looks like our first values for x^(1/3) and x^(2/3) are good.

Our second values are x^(1/3) = .5 and x^(2/3) = .25

Plug into original equation of :

20x^(2/3) - 6x^(1/3) - 2 = 0 to get:

20*.25 - 6*.5 - 2 = 0
simplify to get:
5 - 3 - 2 = 0
combine like terms to get:
0 = 0

Looks like our second value for x^(1/3) and x^(2/3) are also good.

Your answer are that:

x^(1/3) = -.2
and:
x^(1/3) = .5

To solve for x, you would need to cube both sides of this equation to get:

x = (-.2)^3 = -.0008
and:
x = (.5)^3 = .125

You can use your calculator to prove that these values are good.

Your original equation is:

20x^(2/3) - 6x^(1/3) - 2 = 0

If x = -.008, then this equation becomes:

20*(-.008)^(2/3) - 6*(-.008)^(1/3) - 2 = 0

This becomes:

20* ((-.008)^2)^(1/3) - 6*(-.008)^(1/3) - 2 = 0

-.008^2 = .000064^(1/3) = .04
-.008^(1/3) = -.2

Equation becomes 20 * .04 - 6 * (-.2) - 2 = 0

This becomes .8 - (-1.2) - 2 = 0

This becomes .8 + 1.2 - 2 = 0

This becomes 2 - 2 = 0 which is true so the first value for x is good.

I'll leave the confirmation of the second value of x to you.

You're right.

It was a bear of a problem to solve.

The key was understanding that y = x^(1/3) and that y^2 equal x^(2/3).

This was one of the properties of exponents.

(x^a)^2 = x^a * x^a = x^(a+a) = x^(2a)

Even knowing this, the problem was difficult.

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Let x = a positive number.

Then -x = a negative number.

log(b) < 0 implies that log(b) = -x because -x < 0.

Now log(b) = -x if and only if 10^(-x) = b.

This is by basic definition of logarithms.

10^-x is the same as 1/10^x by definition.

Since x >= 0 by definition, then the smallest 10^x could be would be 1 because 10^0 = 1.

Any other value of x > 0 would result in 10^x being greater than 1.

Example:

10^0.1 = 1.2589.....
10^0.00001 = 1.0000023026

Bottom Line is the smallest 10^x can be is 1.

Now, if 1/10^x = b, this means that the largest b can be is 1 because 1/1 = 1.

so, to answer your question:

If log(b) < 0, this means that 0 < b < 1

Some examples:

log(2) = .3...
log(1) = 0
log(.9) = -.04...
log(.5) = -.301...
log(.1) = -1
log(0) = Error - can only take log of a number > 0

So that's the answer to your question.

log(b) < 0 if and only if b is greater than 0 and b is smaller than 1.

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Check out the following hyperlink. It's the most direct and to the point reference I could find.

Pythagoren Triples

This set of formulas should do the trick.

Here's a copy of what's on that website.

There is a simple formula that gives all the Pythagorean triples.
Suppose that m and n are two positive integers, with m < n. Then n2 - m2, 2mn, and n2 + m2 is a Pythagorean triple.
Given any two numbers, m and n, with m < n, you can generate a set of pythagorean triples from it.

example:

3,4

first number is n^2 - m^2 = 16 - 9 = 7
second number is 2mn = 2*3*4 = 24
third number is n^2 + m^2 = 9 + 16 = 25

should be a reight triangle with f^2 + s^2 = t^2

7^2 = 49
24^2 = 576
25^2 = 625

49 + 576 = 625

seems to work.

try m = 1 and n = 2

f (first) = 2^2 - 1^2 = 4-1 = 3
s (second) = 2mn = 2*1*2 = 4
t (third) = 2^2 + 1^21 = 4+1 = 5

not too shabby.

try 5 and 7

f = 7^2 - 5^2 = 49 - 25 = 24
s = 2*5*7 = 10*7 = 70
t = 7^2 + 5^2 = 49 + 25 = 74
24^2 + 70^2 = 74^2 becomes:
576 + 4900 = 5476 becomes:
5476 = 5476

It works !!!!

A pythagorean triple is a set of integers that forms a right triangle.

This formula appears to work in all cases.

Here is a much more complicated explanation which winds up with the same formula stated above.

Pythagorean Triples Formuls and Examples

Here's a definition.

Definition of Pythagorean Triple

the derivtion gets pretty complicated, but the basic formula as stated in the first hyperlink holds.

You can find lots more info by doing a search on the web for "pythagorean triples"

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Scale of the map is 1:50,000

Unless otherwise specified, this is usually a linear measurement.

if it's 1 cm, then real life would be 50,000 cm.

1cm * 1 cm = 1 cm^2 on the map.

50,000 cm * 50,000 cm = 2,500,000,000 cm^2 in real life.

What you would get is 1 cm^2 on the map equals 2,500,000,000 cm^2 in real life.

This means that a farm with an area of 6 cm^2 on the map would be equivalent to a farm with an area of 6*2,500,000,000 cm^2 in real life which would be equal to 15,000,000,000 cm^2 in real life.

Now 1 centimeter = 1/100 of a meter = .01 meter.

1 cm * 1 cm = 1 cm^2

.01 m * .01 m = .0001 m^2

You have 1 cm^2 = .0001 m^2

15,000,000,000 cm^2 * .0001 = 1,500,000 m^2

Since 1 hectare = 10,000 m^2, this means that 1 m^2 = 1/10,000 of a hectare = .0001 hectares.

1,500,000 m^2 * .0001 = 150 hectares.

I believe your answer will be 150 hectares. *****************************

The arithmetic looks good.

If the scale of the map means what I think it does, then these measurements should be good and he area of the farm in hectares should be 150.

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g(x) = 2x - 5 is the equation of a line and is a function because there is only one value of y for each value of x.

The inverse function of g(x) is found by:

Solving for x.
Replacing x with y and y with x.

Let y = g(x)

Your equation becomes y = 2x - 5

Solve for y.

subtract y from both sides of the equation to get:
0 = 2x - 5 - y
subtract 2x from both sides of the equation to get:
-2x = -y - 5
divide both sides of the equation by -2 to get:
x = y/2 + 5/2

Replace x with y and y with x to get:

y = x/2 + 5/2

If this is the inverse function, then both equations will be a reflection about the line y = x.

A graph of these equations and the equation of y = x is shown below:



If these are inverse functions, then:

f(g(x) = g(f(x)

Take f(g(x))

f(x) = 2x-5
g(x) = x/2 + 5/2

f(g(x) = f(x/2+5/2) = 2 * (x/2 + 5/2) - 5 = x + 5 - 5 = x

g(f(x) = g(2x-5) = (2x-5)/2 + 5/2 = x - 5/2 + 5/2 = x

We have f(g(x) = g(f(x) which confirms that these equations are inverse functions of each other.

Definition of a function states that you have a 1 to 1 mapping form x to y.

This happens with both these equations as shown in the graph.

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When you change the signs of an inequality, then the inequality reverses.

a > b becomes -a < -b when you change the signs.
a = b remains -a = -b when you change the signs.
a < b becomes -a > -b when you change the signs.

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this looks like it addresses your problem Expected Value

In your problem, the probability and the payoffs are as follows:

p(red) = 3/10 = .3
p(blue) = 4/10 = .4
p(yellow) = 2/10 = .2
p(green) = 1/10 = .1

Sum of all probabilities equals 1 which it should.

$4.00 payoff for red * .3 = $1.20
$2.00 payoff for green * .1 = $.20
$2.00 payback for blue * .4 = -$.80
$3.00 payback for yellow * .2 = -$.60

Expected Value would be the sum of these = 1.2 + .2 - .8 - .6 = 1.4 - 1.4 = 0

The spinner can expect to break even in the long run.

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106! / 104! is the same as 106*105*104! / 104!.

The 104! in the numerator and the denominator cancels out and you are left with:

106 * 105 = 11130

Take smaller numbers to prove this is good.

Take 5! / 3!

Using your calculator, you would get 20

Doing 5*4*3!/3! you get 5*4 = 20

Same answer.

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you can solve y = 2^x easily enough.
In order to solve x = 2^y, you would have to solve for y.
x = 2^y if and only if y =

for example:

2^3 = 8

should equal 3

Convert to base of 10 in order to solve this using your calcultor.

Use the conversion formula of =

This becomes = 3.

To graph these equations, therefore:

y = 2^x can be graphed directly.

x = 2^y can be graphed by solving for y and getting y =

A graph of both of these equations looks like:



Since both these equations are reflections about the line y = x, this means that y = is the inverse function of y = 2^x.

The same graph with the line y = x added shows this.



In fact, in order to find the inverse function of y = 2^x, you would have to solve for x and then replace x with y and y with x.

You would wind up with the same equation.

y = 2^x if and only if x =

So when you solve for x, you get x =

When you replace y with x and x with y, you get:

y = which is the inverse equation of y = 2^x and is exactly the equation we got when we solved for y in order to be able to graph it.