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Start with the given expression.


Rewrite the logs using the identity


Convert to radical notation.


Combine the first two logs using the identity


Combine the logs using the identity


So

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Simply find the probability of 2 women or 3 women. In other words,


P(at least 2 women) = P(2 women) + P(3 women)

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The locus of points in a plane that are equidistant from points A and B in the plane is a line that is the perpendicular bisector to line that connects the two points.

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Start with the given equation.


Subtract from both sides.


Rearrange the terms.


Divide both sides by to isolate y.


Break up the fraction.


Reduce.

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This question has been answered here.

Edit:

From the previous solution, the given equation is


Start with the given equation.


Distribute


Multiply both sides by 100 to clear out the decimals.


Distribute and multiply.


Subtract from both sides.


Subtract from both sides.


Combine like terms on the left side.


Combine like terms on the right side.


Divide both sides by to isolate .


Reduce.


----------------------------------------------------------------------

Answer:

So the solution is which means that you need 2 cubic feet of soil containing 45% sand.

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This question has already been answered here.

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Let time of express train and time of local train

Also, let speed of express train and speed of local train


Since the "express train travels twice as fast as the local train", this means that


Also, because "the express train arrives 2 hours ahead of the local train", this tells us



Now let's set up the formula for the local train:


Start with the distance rate time formula


Plug in , and


Divide both sides by .


So after isolating the second speed, we get



Now let's set up the formula for the express train:


Start with the distance rate time formula


Plug in , and


Plug in and


Plug in


Multiply


Multiply both sides by .


Distribute


Subtract from both sides.


Combine like terms.


Divide both sides by -60 to isolate .


Divide


So it takes 4 hours for the local train


Go back to the local train formula


Plug in


Divide both sides by 4.


Divide.


So the speed of the local train is 15 mph

Now simply double the speed of the local train to get

So the speed of the local train is 15 mph and the speed of the express train is 30 mph.



Note: nowhere in the solution above is a mention of 12:00 noon. This information is unnecessary and it is put in there to throw you off.

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Start with the given expression.


Factor the numerator


Factor the denominator


Factor 4 to get 2*2


Cancel out the common terms.


Simplify


Expand and multiply


So where

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If we just focus on one triangle, we get




To find the unknown length, we need to use the Pythagorean Theorem.


Remember, the Pythagorean Theorem is where "a" and "b" are the legs of a triangle and "c" is the hypotenuse.


Since the legs are and (ie the legs are equal) this means that and


Also, since the hypotenuse is , this means that .


Start with the Pythagorean theorem.


Plug in , ,


Square to get .


Combine like terms.


Divide both sides by 2.


Reduce


Take the square root of both sides. Note: only the positive square root is considered (since a negative length doesn't make sense).


Evaluate the square root of 36 to get 6


So the side length of the square is 6 units.


Now simply plug this side length into the following formulas (note: "s" is the length of the side of the square):


Perimeter: . So the perimeter is 24 units.

Area: . So the area is 36 square units.

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x-intercept

To find the x-intercept, plug in and solve for x


Start with the given equation.


Plug in .


Subtract 4x from both sides.


Divide both sides by to isolate .


Reduce.


So the x-intercept is .


------------------------------------------

y-intercept

To find the y-intercept, plug in and solve for y


Start with the given equation.


Plug in .


Multiply and 0 to get 0.


Simplify.


So the y-intercept is .

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x-intercept

To find the x-intercept, plug in and solve for x


Start with the given equation.


Plug in .


Simplify.


Divide both sides by to isolate .


Reduce.


So the x-intercept is .

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let x= # of pounds of soybean meal and y=# of pounds of cornmeal

Since we want a 280 lbs mixture, this means that we have the first equation:



Also, because we want a "280-lb mixture that is 12% protein" from the mixture of
soybean meal that is 14% protein and cornmeal that is 7% protein, we have the second equation:



Start with the second equation.


Multiply


Multiply both sides by 100 to remove the decimals


Distribute


So we have the new system of equations:




Let's solve the system by use of substitution


Now in order to solve this system by using substitution, we need to solve (or isolate) one variable. I'm going to solve for y.




So let's isolate y in the first equation


Start with the first equation


Subtract from both sides


Rearrange the equation


---------------------

Since , we can now replace each in the second equation with to solve for



Plug in into the second equation. In other words, replace each with . Notice we've eliminated the variables. So we now have a simple equation with one unknown.



Distribute to


Multiply


Combine like terms on the left side


Subtract 1960 from both sides


Combine like terms on the right side


Divide both sides by 7 to isolate x


Divide



-----------------First Answer------------------------------


So the first part of our answer is:



Since we know that we can plug it into the equation (remember we previously solved for in the first equation).



Start with the equation where was previously isolated.


Plug in


Multiply


Combine like terms



-----------------Second Answer------------------------------


So the second part of our answer is:






-----------------Summary------------------------------

So our answers are:

and which form the ordered pair (200,80)



So 200 pounds of soybean meal and 80 pounds of cornmeal are needed for the mixture.

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x-intercept

To find the x-intercept, plug in and solve for x


Start with the given equation.


Plug in .


Multiply and 0 to get 0.


Simplify.


Divide both sides by to isolate .


Reduce.


So the x-intercept is .


------------------------------------------

y-intercept

To find the y-intercept, plug in and solve for y


Start with the given equation.


Plug in .


Multiply and 0 to get 0.


Simplify.


Divide both sides by to isolate .


Reduce.


So the y-intercept is .


Notice that if we graph the equation , we can see that the graph has the x-intercept and the y-intercept


Graph of

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Note: is the first point and is the second point .


Start with the slope formula.


Plug in , , , and


Subtract from to get


Subtract from to get


Remember, you cannot divide by zero. So this means that the slope is undefined.

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Start with the given equation.


Add 5 to both sides.


Subtract from both sides.


Rearrange the terms.


Divide both sides by to isolate y.


Break up the fraction.


Reduce.


So the equation is now in slope intercept form where the slope is and the y-intercept is


note: the y-intercept is the point

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Start with the given equation.


Subtract from both sides.


Rearrange the terms.


Divide both sides by to isolate y.


Break up the fraction.


Reduce.


So the equation is now in slope intercept form where the slope is and the y-intercept is

note: the y-intercept is the point

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No matter how many leaflets you buy, you will pay at least £100. So that's why the constant cost of 100 is in there. Any leaflets you buy, you'll be paying 0.05 (or 5%) of a pound per leaflet. So this is why 0.05N is added.


Here are some examples:

If you buy 100 leaflets, then N=100 and...

C = 100 + 0.05(100) = 100 + 5 = 105

So the total cost is £105



Or if you buy 300 leaflets, then N=300 and...

C = 100 + 0.05(300) = 100 + 15 = 115

So the total cost is £115

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a)

Since you cannot take the log of 0 or a negative value, this means that and (for the outer and inner logs)


If we solve , we get . So combining the inequalities and (ie perform a set union), we get .

So the domain of is


------------------

b)

Start with the given function.


Replace f(x) with "y"


Switch each x and y


Rewrite the equation using the property: ====>


Rewrite the equation again using the property: ====>


So the inverse function is

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Data Set #1: 1, 5, 33, 36

Range = 36 - 1 = 35
Standard Deviation is relatively large (since the data values are spread out)



Data Set #2: 1, 2, 4, 7, 9, 12, 13, 14, 16, 19, 20, 21, 25, 27, 29, 30, 31, 34, 35, 36

Range = 36 - 1 = 35
Now the standard deviation is slightly smaller (since the data values are closer together)

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Simply subtract 2 times the standard deviation from the mean to get:


58-2*6.3=58-12.6=45.4


So the value that is 2 standard deviations below the mean is 45.4

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If you either draw a picture or hold up two sheets of paper (for the two planes), you'll see that the locus of points form a third plane that is right in the middle of the two parallel planes.

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1. Integers: any whole number
Let a=2, b=6, c=-12 (which are all whole numbers)



2. Rational Number: any number that is a ratio (or fraction) of two integers. So and are rational numbers
Let ,, (which are all fractions of whole numbers)



3. Irrational number: any number that cannot be represented by a fraction of two integers. For example and are irrational numbers

, , ,

,

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The function is a quadratic polynomial function.

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This question has been answered here. Did you have any questions about it?

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First sort the numbers into ascending order (from least to greatest):
0.2,0.2,0.3,0.4,0.6,0.6,0.9,1.1,1.2,1.3,1.4


Range:

To find the range, subtract the smallest number (in this case 0.2) from the largest number (in this case 1.4) like this:




So the range is 1.2

---

Interquartile Range:


First, find the median for the first half of the data set (ignore the median):


Median of 0.2, 0.2, 0.3, 0.4, 0.6 is 0.3 (since this is the middle most value)


Now find the median of the last half of the data set (ignore the median):


Median of 0.9, 1.1, 1.2, 1.3, 1.4 is 1.2 (since this is the middle most value)


So the 1st and 3rd quartiles are 0.3 and 1.2 respectively.


This means that the interquartile range is

1.2 - 0.3 = 0.9


Note: the interquartile range is the difference between the 1st and 3rd quartiles

---

Note: I'm assuming that you want the sample standard deviation.


Standard Deviation:


Step 1) Find the mean:






So the mean is roughly 0.75 (rounded to the nearest hundredth).


----------------------------------------------------------


Step 2) Find the differences between the mean and every value:


Now subtract EVERY value from the mean (0.75) to get:
0.2-0.75 = -0.55, 0.2-0.75 = -0.55, 0.3-0.75 = -0.45, 0.4-0.75 = -0.35, 0.6-0.75 = -0.15, 0.6-0.75 = -0.15, 0.9-0.75 = 0.15, 1.1-0.75 = 0.35, 1.2-0.75 = 0.45, 1.3-0.75 = 0.55, 1.4-0.75 = 0.65,


So you should have a new column of values:
-0.55, -0.55, -0.45, -0.35, -0.15, -0.15, 0.15, 0.35, 0.45, 0.55, 0.65,
----------------------------------------------------------


Step 3) Squaring the differences:


Now square EVERY value in the last column to get:
(-0.55)^2 = 0.3, (-0.55)^2 = 0.3, (-0.45)^2 = 0.2, (-0.35)^2 = 0.12, (-0.15)^2 = 0.02, (-0.15)^2 = 0.02, (0.15)^2 = 0.02, (0.35)^2 = 0.12, (0.45)^2 = 0.2, (0.55)^2 = 0.3, (0.65)^2 = 0.42,


So you should have a new column of values:
0.3, 0.3, 0.2, 0.12, 0.02, 0.02, 0.02, 0.12, 0.2, 0.3, 0.42,
----------------------------------------------------------


Step 4) Adding up the squares:


Now add up the values of the last column:
0.3 + 0.3 + 0.2 + 0.12 + 0.02 + 0.02 + 0.02 + 0.12 + 0.2 + 0.3 + 0.42 = 2.02

----------------------------------------------------------

Step 5) Divide the sum by n-1:


Divide that sum by one less than the number of values in the list (in this case, it is n-1=11-1=10).





----------------------------------------------------------


Step 6) Take the square root of the quotient:


Take the square root of to get


Now round to the nearest hundredth to get .


So the standard deviation is roughly .

---

If you don't like that method, then here's another way:


First sort the numbers into ascending order (from least to greatest):
0.2,0.2,0.3,0.4,0.6,0.6,0.9,1.1,1.2,1.3,1.4


Standard Deviation: where is the average, is the ith number, and is the number of numbers

So we can replace N with 11



Subtract to get 10



Replace with 0.75



Expand the summation (replace each with the respective number)




Subtract the terms in the parenthesis



Square each term



Add up all of the terms



Multiply



Take the square root




Round to the nearest hundredth




So the standard deviation is roughly

---

25th and 75th percentiles:

The 25th and 75th percentiles are simply the first and third quartiles. We found these earlier to be 0.3 and 1.2.



So the 25th and 75th percentiles are 0.3 and 1.2

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Let's say that you want to find the distance between two points. You will then use that distance and add it to other distances you find. Also, you want to keep your calculations as exact as possible (to avoid any errors). To keep things exact, you most likely end up with a radical.

For example, let's say that you want to find the distance between the points (1,2) and (5,8):

Start with the distance formula.


Plug in , , , and .


Subtract from to get .


Subtract from to get .


Square to get .


Square to get .


Add to to get .


Simplify the square root.


Now if you found other distances (in terms of radicals), you could add them to this given distance and keep things exact. Once you are done with all of your calculations, you can use a calculator to find the approximate value.

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# 3

"If tomorrow is not the day to go shopping, then today is not Friday" translates into the symbolic argument: q -> ~p

Recall, ~p means NOT p and -> means "if..then.."

# 4
What is the symbol "Ù". Is this a typo?

# 5

Original Statement: If the sun is shining, then it will not rain.
Converse: If it will not rain, then the sun is shining
Inverse: If the sun is not shining, then it will rain
Contrapositive: If it will rain, then the sun is not shining

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Start with the given expression.


Now let's FOIL the expression.


Remember, when you FOIL an expression, you follow this procedure:


Multiply the First terms:.


Multiply the Outer terms:.


Multiply the Inner terms:.


Multiply the Last terms:.


---------------------------------------------------
So we have the terms: , , , and


Now add every term listed above to make a single expression.


Now combine like terms.


So FOILs to .


In other words, .

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Recall that . If you have a negative exponent, then you simply flip the fraction to make it positive.

So


which means that


------------------------------------------------

Also, we'll use the identity


Start with the given expression.


Use the identity given above to multiply the outer exponent with EVERY exponent inside the parenthesis.


Multiply


Now flip the fraction to make the exponent positive


Multiply


So

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You are correct. Good job.

You can check it by letting "b" be some easy number to deal with like 100. So 0.35*100=35. Since 35% of a 100 is 35, this verifies the expression.

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Start with the given equation.


Subtract from both sides.


Rearrange the terms.


Divide both sides by to isolate y.


Break up the fraction.


Reduce.


We can see that the equation has a slope and a y-intercept .


Now to find the slope of the perpendicular line, simply flip the slope to get . Now change the sign to get . So the perpendicular slope is .


Now let's use the point slope formula to find the equation of the perpendicular line by plugging in the slope and the coordinates of the given point .


Start with the point slope formula


Plug in , , and


Rewrite as


Distribute


Multiply


Add 8 to both sides.


Combine like terms.


So the equation of the line perpendicular to that goes through the point is .


The equation in function notation would then be


Here's a graph to visually verify our answer:
Graph of the original equation (red) and the perpendicular line (green) through the point .