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 Linear-systems/148200: You are sliding a big cabinet down a truck ramp for a customer. You must maintain control of the cabinet because it is very heavy. There are 2 ramps in the building. The equation of ramp 1 is y = 2x + 7 and the equation of ramp 2 is y = 3 x + 7. Which is the safest ramp to slide the cabinet down without losing control and why? 1 solutions Answer 108594 by jim_thompson5910(28717)   on 2008-07-14 11:34:33 (Show Source): You can put this solution on YOUR website!Notice how the slope of the first equation is and the the slope of the second equation is . Since the slope of the second equation is larger than the the slope of the first equation, this tells us that the second ramp is steeper. So the first ramp is safer since it has a more shallow slope.
 Functions/148198: Find the range of h(x)=2-sqrt(x-1)1 solutions Answer 108592 by jim_thompson5910(28717)   on 2008-07-14 11:20:40 (Show Source): You can put this solution on YOUR website!When you graph , you get From the graph, we can see that the highest point is at (1,2). So this means that the highest y value is . So this means that y can be any number less than or equal to 2. So this tells us that the range is So the range in set-builder notation is Also the range in interval notation is (]
 Linear-equations/148196: 20) Find the slope and the y-intercept. y = 4x + 2 Can someone please help. v/r Jess1 solutions Answer 108591 by jim_thompson5910(28717)   on 2008-07-14 11:18:31 (Show Source): You can put this solution on YOUR website! Notice how is in slope-intercept form where m is the slope and b is the y-intercept. So this shows us that the slope is and the y-intercept is which is the point (0,2)
 Polynomials-and-rational-expressions/148170: f(x)=x^5-3x^4-3x^3+9x^2-4x+12 How do i find only the rational zeros for this equation? 1 solutions Answer 108562 by jim_thompson5910(28717)   on 2008-07-14 01:10:47 (Show Source): You can put this solution on YOUR website!Any rational zero can be found through this equation where p and q are the factors of the last and first coefficients So let's list the factors of 12 (the last coefficient): Now let's list the factors of 1 (the first coefficient): Now let's divide each factor of the last coefficient by each factor of the first coefficient Now simplify These are all the distinct rational zeros of the function that could occur
Polynomials-and-rational-expressions/148168: (x^3-7x^2+13x+3) ÷ (x-2)
I'm having difficulty with synthetic division. How do i find the quotient and the remainder in this problem? Can you also be explicit in the steps please.
1 solutions

Answer 108561 by jim_thompson5910(28717)   on 2008-07-14 00:08:00 (Show Source):
You can put this solution on YOUR website!

Let's simplify this expression using synthetic division

First lets find our test zero:

Set the denominator equal to zero

Solve for x.

so our test zero is 2

Now set up the synthetic division table by placing the test zero in the upper left corner and placing the coefficients of the numerator to the right of the test zero.
 2 | 1 -7 13 3 |

Start by bringing down the leading coefficient (it is the coefficient with the highest exponent which is 1)
 2 | 1 -7 13 3 | 1

Multiply 2 by 1 and place the product (which is 2) right underneath the second coefficient (which is -7)
 2 | 1 -7 13 3 | 2 1

Add 2 and -7 to get -5. Place the sum right underneath 2.
 2 | 1 -7 13 3 | 2 1 -5

Multiply 2 by -5 and place the product (which is -10) right underneath the third coefficient (which is 13)
 2 | 1 -7 13 3 | 2 -10 1 -5

Add -10 and 13 to get 3. Place the sum right underneath -10.
 2 | 1 -7 13 3 | 2 -10 1 -5 3

Multiply 2 by 3 and place the product (which is 6) right underneath the fourth coefficient (which is 3)
 2 | 1 -7 13 3 | 2 -10 6 1 -5 3

Add 6 and 3 to get 9. Place the sum right underneath 6.
 2 | 1 -7 13 3 | 2 -10 6 1 -5 3 9

Since the last column adds to 9, we have a remainder of 9. This means is not a factor of
Now lets look at the bottom row of coefficients:

The first 3 coefficients (1,-5,3) form the quotient

and the last coefficient 9, is the remainder, which is placed over like this

Putting this altogether, we get:

So

which looks like this in remainder form:
remainder 9

You can use this online polynomial division calculator to check your work

Polynomials-and-rational-expressions/148141: how do i factor 20c(squared+17c-63r squared

1 solutions

Answer 108559 by jim_thompson5910(28717)   on 2008-07-13 23:29:08 (Show Source):
You can put this solution on YOUR website!

Looking at we can see that the first term is and the last term is where the coefficients are 20 and -63 respectively.

Now multiply the first coefficient 20 and the last coefficient -63 to get -1260. Now what two numbers multiply to -1260 and add to the middle coefficient 17? Let's list all of the factors of -1260:

Factors of -1260:
1,2,3,4,5,6,7,9,10,12,14,15,18,20,21,28,30,35,36,42,45,60,63,70,84,90,105,126,140,180,210,252,315,420,630,1260

-1,-2,-3,-4,-5,-6,-7,-9,-10,-12,-14,-15,-18,-20,-21,-28,-30,-35,-36,-42,-45,-60,-63,-70,-84,-90,-105,-126,-140,-180,-210,-252,-315,-420,-630,-1260 ...List the negative factors as well. This will allow us to find all possible combinations

These factors pair up and multiply to -1260
(1)*(-1260)
(2)*(-630)
(3)*(-420)
(4)*(-315)
(5)*(-252)
(6)*(-210)
(7)*(-180)
(9)*(-140)
(10)*(-126)
(12)*(-105)
(14)*(-90)
(15)*(-84)
(18)*(-70)
(20)*(-63)
(21)*(-60)
(28)*(-45)
(30)*(-42)
(35)*(-36)
(-1)*(1260)
(-2)*(630)
(-3)*(420)
(-4)*(315)
(-5)*(252)
(-6)*(210)
(-7)*(180)
(-9)*(140)
(-10)*(126)
(-12)*(105)
(-14)*(90)
(-15)*(84)
(-18)*(70)
(-20)*(63)
(-21)*(60)
(-28)*(45)
(-30)*(42)
(-35)*(36)

note: remember, the product of a negative and a positive number is a negative number

Now which of these pairs add to 17? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 17

First NumberSecond NumberSum
1-12601+(-1260)=-1259
2-6302+(-630)=-628
3-4203+(-420)=-417
4-3154+(-315)=-311
5-2525+(-252)=-247
6-2106+(-210)=-204
7-1807+(-180)=-173
9-1409+(-140)=-131
10-12610+(-126)=-116
12-10512+(-105)=-93
14-9014+(-90)=-76
15-8415+(-84)=-69
18-7018+(-70)=-52
20-6320+(-63)=-43
21-6021+(-60)=-39
28-4528+(-45)=-17
30-4230+(-42)=-12
35-3635+(-36)=-1
-11260-1+1260=1259
-2630-2+630=628
-3420-3+420=417
-4315-4+315=311
-5252-5+252=247
-6210-6+210=204
-7180-7+180=173
-9140-9+140=131
-10126-10+126=116
-12105-12+105=93
-1490-14+90=76
-1584-15+84=69
-1870-18+70=52
-2063-20+63=43
-2160-21+60=39
-2845-28+45=17
-3042-30+42=12
-3536-35+36=1

From this list we can see that -28 and 45 add up to 17 and multiply to -1260

Now looking at the expression , replace with (notice adds up to . So it is equivalent to )

Now let's factor by grouping:

Group like terms

Factor out the GCF of out of the first group. Factor out the GCF of out of the second group

So this also means that factors to (since is equivalent to )

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

So factors to

 Linear-equations/148143: I don't understand slope and m x +b and all that... how do you solve it because i have the TAkZ test comiN up on wedsnday! PLz1 solutions Answer 108558 by jim_thompson5910(28717)   on 2008-07-13 23:27:47 (Show Source): You can put this solution on YOUR website!Do you want graph a linear equation? Let's say that you want to graph Looking at we can see that the equation is in slope-intercept form where the slope is and the y-intercept is Since this tells us that the y-intercept is .Remember the y-intercept is the point where the graph intersects with the y-axis So we have one point Now since the slope is comprised of the "rise" over the "run" this means Also, because the slope is , this means: which shows us that the rise is 1 and the run is 2. This means that to go from point to point, we can go up 1 and over 2 So starting at , go up 1 unit and to the right 2 units to get to the next point Now draw a line through these points to graph So this is the graph of through the points and
 Systems-of-equations/148146: Hi, Please Solve the systems of equations. 2x+Y=3 4x+3Y=1 Thank you!1 solutions Answer 108557 by jim_thompson5910(28717)   on 2008-07-13 23:26:01 (Show Source): You can put this solution on YOUR website! Start with the given system of equations: Multiply the both sides of the first equation by -2. Distribute and multiply. So we have the new system of equations: Now add the equations together. You can do this by simply adding the two left sides and the two right sides separately like this: Group like terms. Combine like terms. Notice how the x terms cancel out. Simplify. ------------------------------------------------------------------ Now go back to the first equation. Plug in . Multiply. Subtract from both sides. Combine like terms on the right side. Divide both sides by to isolate . Reduce. So our answer is and . Which form the ordered pair . This means that the system is consistent and independent. Notice when we graph the equations, we see that they intersect at . So this visually verifies our answer. Graph of (red) and (green)
 Functions/148163: Simplify 1 solutions Answer 108553 by jim_thompson5910(28717)   on 2008-07-13 22:13:05 (Show Source): You can put this solution on YOUR website! Start with the given expression. Factor 81 into 9*9 Break up the root. Factor out the GCF Factor 9 into 3*3 Break up the root. Highlight the common terms. Cancel out the common terms. Simplify Distribute Combine the roots and multiply So simplifies to In other words,
 Functions/148161: f(x)=1-x^2 and g(x)=1-x. Find (f/g)(x)1 solutions Answer 108549 by jim_thompson5910(28717)   on 2008-07-13 21:37:18 (Show Source): You can put this solution on YOUR website! Start with the given property Plug in and Factor the numerator. Highlight the common terms. Cancel out the common terms. Simplify Rearrange the terms.
 Functions/148159: Simplify (v^25/w^15)^(3/5)1 solutions Answer 108548 by jim_thompson5910(28717)   on 2008-07-13 21:25:08 (Show Source): You can put this solution on YOUR website! Start with the given expression. Distribute the exponent. Multiply the exponents. Reduce So simplifies to . In other words, where every variable is positive and
 Radicals/148158: Simplify 1 solutions Answer 108545 by jim_thompson5910(28717)   on 2008-07-13 21:18:39 (Show Source): You can put this solution on YOUR website! Start with the given expression. Factor 81 into 9*9 Break up the root. Factor out the GCF Factor 9 into 3*3 Break up the root. Highlight the common terms. Cancel out the common terms. Simplify Distribute Combine the roots and multiply So simplifies to In other words,
 Functions/148157: f(x)=(3-x)(3+x) and g(x)=3-x. Find (f/g)(x)1 solutions Answer 108543 by jim_thompson5910(28717)   on 2008-07-13 21:06:37 (Show Source): You can put this solution on YOUR website! Start with the given property Plug in and Highlight the common terms. Cancel out the common terms. Simplify. Rearrange the terms.
 Trigonometry-basics/148154: 2. Out for a walk in Chicago, Morgan measured the angle of elevation to the distant Sears Tower, and got 3.6 degrees. After walking one km directly toward the building, Morgan found that the angle of elevation had increased to 4.2 degrees. Use this information to estimate the height of the Sears Tower and how far Morgan is from it after walking toward the building. 1 solutions Answer 108537 by jim_thompson5910(28717)   on 2008-07-13 20:33:28 (Show Source): You can put this solution on YOUR website!First draw the picture and label the angles and sides: Now let's find the measures of the other angles. From the drawing, we can see that the angle 4.2 degrees and the unkown angle "a" make up a straight line. So this means that . Solving for "a", we get . Using the fact that all angles in a triangle add to 180 degrees, this means that (these are the angles from the obtuse triangle). Solving for "b" gives us . Finally, we know that angles "c" and 4.2 are complementary. So this means that . Isolate "c" and we get Now add these angle measures to the drawing. Notice how we have a lot of angle measures but only one side length given. We can use the Law of Sines to find any given length of any triangle as long as we have at least two angles and a side length. So the Law of Sines formula is: Since we have two angles and a side, let's use the angle measures 3.6 and 0.6 to find the length of "y" Plug in , , , and Simplify. Multiply both sides by . Divide both sides by . Rearrange the equation. Evaluate the sine functions. Divide. So the length of "y" is 5.99616 km Now let's update the drawing From the drawing, we can see that the hypotenuse of the second triangle is 5.99616. Using the angle 4.2, this makes the height of the building "x" the opposite side. So let's use the sine function where "a" is the angle 4.2 Plug in and the given side lengths Multiply both sides by 5.99616. Evaluate the sine of 4.2. Multiply So the height is . Now to find the length of the base of the second triangle, simply use the cosine function. where "a" is the angle 4.2 Plug in and the given side lengths Multiply both sides by 5.99616. Evaluate the cosine of 4.2. Multiply So the distance from the building to the second position is 5.97997 km --------------------------------- Answer: So the height of the building is 0.43915 km or 1,440.78084 feet and the distance from the building to Morgan's final position is 5.97997 km or 19,619.3241 feet
 Trigonometry-basics/148137: 13. Atiba wants to measure the width of the Hudson River. Standing under a tree T on the river bank, Atiba sights a rock at the nearest point R on the opposite bank. Then Atiba walks to a point P on the river bank that is 50 meters from T, and makes RTP a right angle. Atiba then measures RPT and obtains 76.8 degrees. How wide is the river?1 solutions Answer 108534 by jim_thompson5910(28717)   on 2008-07-13 19:39:09 (Show Source): You can put this solution on YOUR website!First let's draw a picture of the problem: From the picture, we can see that the opposite side is "x" and the adjacent side is 50. So let's use the tangent function to find the unknown length Remember, the tangent function is where "a" is the angle Start with the given equation. Plug in and the given lengths of the legs of the triangles. Multiply both sides by 50. Take the tangent of 76.8 to get 4.264 (note: make sure that you are in "degree" mode) Multiply So the answer is approximately which means that the width of the river is about 213.2 meters
 Functions/148130: Hi, looking for the Domain and the Range of Y= the Squareroot of (X-3) Also looking for the inverse of 4x+7. thank you.1 solutions Answer 108533 by jim_thompson5910(28717)   on 2008-07-13 19:25:20 (Show Source): You can put this solution on YOUR website! Start with the given expression Remember you cannot take the square root of a negative value. So that means the argument must be greater than or equal to zero (i.e. the argument must be positive) Set the inner expression greater than or equal to zero Add 3 to both sides Combine like terms on the right side So that means x must be greater than or equal to in order for x to be in the domain So the domain in set-builder notation is So here is the domain in interval notation: [3,) Now to find the range, we must graph the equation Graph of From the graph, we can see that the lowest y-value is . So the range in set-builder notation is Also the range in interval notation is: [0,) Start with the given equation. Switch x and y. Subtract 7 from both sides. Divide both sides by 4. So the inverse function is
 Equations/148129: (v-1)/(v+3) = (v+3)/v v=?1 solutions Answer 108524 by jim_thompson5910(28717)   on 2008-07-13 18:50:07 (Show Source): You can put this solution on YOUR website! Start with the given expression Cross multiply Distribute Foil Add v to both sides. Subtract from both sides. Combine like terms. Subtract 9 from both sides. Divide both sides by 7. So the answer is
 Graphs/148127: Simplify 1 solutions Answer 108516 by jim_thompson5910(28717)   on 2008-07-13 18:18:35 (Show Source): You can put this solution on YOUR website! Start with the given expression. Factor into . Notice how 8 is a perfect cube (ie ). Factor into . Once again, notice how is a perfect cube. Factor into . Once again, notice how is a perfect cube. I'm breaking the expression into perfect cubes so that when I take the cube root of these cubes, I'll be left with the expression itself. Since is a perfect cube, this means that . Also, since is a perfect cube, this means that . Break up the root. Take the cube root of the perfect cubes to get just the base of the expression. Recombine any roots leftover and multiply So simplifies to In other words, .
 Rational-functions/148124: Simplify 1 solutions Answer 108513 by jim_thompson5910(28717)   on 2008-07-13 18:12:25 (Show Source): You can put this solution on YOUR website! Start with the given expression. Factor into . Notice how 8 is a perfect cube (ie ). Factor into . Once again, notice how is a perfect cube. Factor into . Once again, notice how is a perfect cube. I'm breaking the expression into perfect cubes so that when I take the cube root of these cubes, I'll be left with the expression itself. Since is a perfect cube, this means that . Also, since is a perfect cube, this means that . Break up the root. Take the cube root of the perfect cubes to get just the base of the expression. Recombine any roots leftover and multiply So simplifies to In other words, .
 Rational-functions/148121: Combine  4 --- - x^(1/3) x^(2/3) 1 solutions Answer 108508 by jim_thompson5910(28717)   on 2008-07-13 17:58:02 (Show Source): You can put this solution on YOUR website! Start with the given expression. Notice how the LCD is . So we need to multiply the second term by the LCD . Multiply the second term by the LCD Combine the fractions. Multiply the terms by adding the exponents. Add. Reduce. Simplify. Since the denominators are now the same, we can add the fractions. Combine the numerators over the common denominator. So simplifies to In other words, where x is positive and
Rational-functions/148119: Factor: (2x^(2/3)) - (5x^(1/3)) - 3
1 solutions

Answer 108504 by jim_thompson5910(28717)   on 2008-07-13 17:45:26 (Show Source):
You can put this solution on YOUR website!

Let . This means that

Replace with . Replace with

Looking at we can see that the first term is and the last term is where the coefficients are 2 and -3 respectively.

Now multiply the first coefficient 2 and the last coefficient -3 to get -6. Now what two numbers multiply to -6 and add to the middle coefficient -5? Let's list all of the factors of -6:

Factors of -6:
1,2,3,6

-1,-2,-3,-6 ...List the negative factors as well. This will allow us to find all possible combinations

These factors pair up and multiply to -6
(1)*(-6)
(2)*(-3)
(-1)*(6)
(-2)*(3)

note: remember, the product of a negative and a positive number is a negative number

Now which of these pairs add to -5? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to -5

First NumberSecond NumberSum
1-61+(-6)=-5
2-32+(-3)=-1
-16-1+6=5
-23-2+3=1

From this list we can see that 1 and -6 add up to -5 and multiply to -6

Now looking at the expression , replace with (notice adds up to . So it is equivalent to )

Now let's factor by grouping:

Group like terms

Factor out the GCF of out of the first group. Factor out the GCF of out of the second group

Since we have a common term of , we can combine like terms

So factors to

Now replace "z" with

So factors to

In other words, where every variable is positive.

 Rational-functions/148115: Multiply: ((3x^(2/3)) + 2)^21 solutions Answer 108500 by jim_thompson5910(28717)   on 2008-07-13 17:37:01 (Show Source): You can put this solution on YOUR website! Start with the given expression. Expand. Remember, Foil. Multiply and to get Multiply and to get . Note: this happens twice since the multiplication order does not matter. Multiply and to get . Combine like terms. So FOILs and multiplies to In other words,
 Rational-functions/148114: Simplify:  ((x^(3/2))y^-3)^1/3 -------------------- ((x^(1/2))y^-6)^-1 1 solutions Answer 108498 by jim_thompson5910(28717)   on 2008-07-13 17:24:32 (Show Source): You can put this solution on YOUR website! Start with the given expression. Distribute the outer exponents to the inner exponents. Remember, . So in our case the numerator goes from to and the denominator goes from to Use the technique described above to distribute the exponents. Multiply the exponents. Reduce. Now when we divide monomials, we simply subtract the exponents. So for example Let's apply this technique to the problem: Subtract the exponents. Rewrite as Combine the exponents. Reduce. Simplify Now rewrite as So simplifies to In other words, where every variable is positive and or