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Graphs/310701: When graphing, how do I know if the line should be solid or broken, and also what part should i shade? 1 solutions Answer 222188 by solver91311(16872)   on 2010-06-02 14:51:14 (Show Source): You can put this solution on YOUR website! If your inequality symbol is or then use a broken line for the boundary. This means that points on the line are NOT included in the solution set. If your inequality symbol is or then use a solid line for the boundary. This means that points on the line ARE included in the solution set. Once you have the boundary line graphed, pick any point you like that is NOT ON THE LINE. If the line does not pass through the origin, the origin itself is a very good selection for a test point. Otherwise, pick something with small integer coefficients -- all this so that the arithmetic is easier. Substitute the coordinates of the point you chose into the original inequality. If the result is a true statement, shade in the side of the line that CONTAINS the point you chose. If the result is a false statement, shade in the side of the line that DOES NOT CONTAIN the point you chose. This doesn't go through the origin so, lets try (0,0) Yep, zero is less than 2, so the shading happens on the side of the line containing the origin. Solved by pluggable solver: Plot Any Inequality Graphing function : See? John

Quadratic_Equations/310688: Hello I have a problem solving this parabola question.The path of a cliff diver as he dives into a lake ,is given by this eqaution,y=-(x-10)(SQAURED)+75,where y metres is the diver's height above the water and,x metres is the horisontal distance travelled by the diver.What is the maximum height the diver is above the water? 1 solutions Answer 222183 by solver91311(16872)   on 2010-06-02 14:32:06 (Show Source): You can put this solution on YOUR website! Not sure whether you just need an intuitive approach, a formal algebraic approach, or a calculus approach. Intuitive The largest y can be is 75 and that is when . That is because is always a positive number, unless it is zero, so is always a negative number unless it is zero. Therefore you are always subtracting something from 75 unless , which happens when While we are at it -- 75 meters? That is 246 feet. Hitting the water from that height would be like hitting concrete. I rather think your cliff diver is only going to perform this particular stunt once. Algebraic Expand the binomial expression: A parabola with a negative lead coefficient opens downward, hence the vertex is a maximum. The vertex of a parabola expressed in form has a vertex that occurs at an value of , so: And the value of the function at is: Calculus Using the function definition from the Algebraic discussion: The function will have a maximum where the first derivative is zero and the second derivative is negative. Set it equal to zero which is negative for all in the domain of So there is a maximum at John

Circles/310663: If angle ema is 95 degrees, what is the measure of angle bme? I would be grateful for any assistance with this problem. Cheers! 1 solutions Answer 222170 by solver91311(16872)   on 2010-06-02 13:48:09 (Show Source): You can put this solution on YOUR website! Sorry, but without the picture or a detailed and unambiguous description of the figure, we can't help you. John

Linear-equations/310690: The coordinates of the end points of a segment are given. Find the coordinates of the midpoint of the segment. G(-8,10),H(-4,8) Please show me how to get the answewr 1 solutions Answer 222165 by solver91311(16872)   on 2010-06-02 13:41:18 (Show Source): You can put this solution on YOUR website! Use the mid-point formulae: and Where and are the coordinates of the given points. John

Polynomials-and-rational-expressions/310687: HI,I WOULD LIKE SOM HELP WITH THIS PROBLEM: 1 solutions Answer 222162 by solver91311(16872)   on 2010-06-02 13:37:57 (Show Source): You can put this solution on YOUR website! I would like you to stop shouting. Typing in all caps is the electronic equivalent of shouting and is therefore both annoying and rude. John

Quadratic_Equations/310685: Under what circumstances would prefer using factoring, completing the square and quadratic formula over each other? 1 solutions Answer 222161 by solver91311(16872)   on 2010-06-02 13:36:47 (Show Source): You can put this solution on YOUR website! Factoring is very handy for quadratics that are easily factored. If you don't see the factorization right off, do a quick mental calculation of the Discriminant, that is the part of the quadratic formula under the radical. If you come up with something other than an perfect square, then the quadratic does not have rational factors, i.e. it is time to move on to another technique. Completing the square is valuable in two situations: First, when you are learning the concept in preparation for development of the quadratic formula, and second when you want to transform a general second-degree equation in standard form into vertex form so that you can identify and graph the conic. All other situations call for the quadratic formula. If you forget the quadratic formula, you can always take the time to complete the square on to derive it -- but memorizing the formula is MUCH easier. John

Mixture_Word_Problems/310674: This question was answered but I'm still confused. Can I please have some more help? High school was ten years ago, what little math ability I had is gone. Thanks Again My question is why is the quanity equation multiplied by 20 and not.2. Also when solving the two equations why does the 20a cancel, but not the 550? I know these are probably very simple, but I just don't understand. How many ml of medication A which has a 20% concentration, and medication B which has a 65% concentration should be mixed to form a 550 ml solution that is 30.9% ------------------------------- Equation: Quantity Equation:::::::: a + b = 550 ml Active Ingredient Eq: 0.20a + 0.65b = 0.309*550 ---------------------- Solving using 2 equations and elimination: Multiply thru the 1st equation by 20 Multiply thru the 2nd equation by 100 ---------------------------------------- 20a + 20b = 20*550 20a + 65b = 30.9*550 ---------------------------- Subtract 1st from 2nd and solve for "b": 45b = 10.9*550 b = 133.22 ml (amt. of 65% solution needed in the mixture) ---- Since a + b = 550, a = 416.78 ml (amt. of 20% solution needed in the mix) 1 solutions Answer 222158 by solver91311(16872)   on 2010-06-02 13:23:29 (Show Source): You can put this solution on YOUR website! Actually, you could multiply the Quantity equation by 2, but then you would have to multiply the Active Ingredient equation by 10 instead of 100. As to your other question, let me illustrate it this way. In the first place, I would have used -20 as a multiplier for the Q equation and 100 for the AI equation, thus: Now, you could multiply out the constant terms in the right-hand sides, and that would actually answer your question about "why the 550 doesn't cancel". But let's proceed without that step. Since I used a -20 multiplier for the Q equation, we can actually add the two equations rather than subtracting them. This technique gives you much less chance for a sign error. But rather than just showing the sum of the two, let's use the idea that if and , then to write: Now, and that is why the goes away. , just like you wrote. For the constant term, let's do a little factoring. Factor out the 550: Putting it all back together: And finally, as you stated, John

Geometry_Word_Problems/310676: The perimeter of an equilateral triangle is at most 57 feet. What could the length of a side be? 1 solutions Answer 222155 by solver91311(16872)   on 2010-06-02 13:00:12 (Show Source): You can put this solution on YOUR website! An equilateral triangle has three sides that are the same length. That's what equilateral means: "equi-" means "the same" and "lateral" means "side" If the perimeter is at most 57 whatevers, then the measure of a side can be at most 57 divided by 3 of the same whatevers. John

Rectangles/310677: find the perimeter and area of a rectangle with width 5 centimeters and length 16 centimeters. 1 solutions Answer 222152 by solver91311(16872)   on 2010-06-02 12:56:38 (Show Source): You can put this solution on YOUR website! The perimeter of a rectangle: The area of a rectangle John

Money_Word_Problems/310669: how much money would i have to put in a cd to draw $500 a month joinnow2000@aol.com 1 solutions Answer 222151 by solver91311(16872)   on 2010-06-02 12:55:00 (Show Source): You can put this solution on YOUR website! That would depend a great deal on the annual interest rate and, to a lesser extent, on the compounding frequency, and if you are just considering an investment that will earn that much that will be allowed to remain in the account or if you want to actually draw out $500.00 every month as an annuity payment. Let's presume annual interest expressed as a decimal. Let's assume that the institution actually pays one-twelfth of the annual interest each month and that you want a monthly pay annuity. Let be the principal amount invested in the CD. Then you need: Solving for Right now, June 2, 2010, the best you will be able to do is John

Graphs/310664: Solve the system by graphing: 2x+y=12 2x+y=20 1 solutions Answer 222144 by solver91311(16872)   on 2010-06-02 12:25:07 (Show Source): You can put this solution on YOUR website! Two parallel lines, no solution. How did I know? If you have: and and and where Then you have two parallel lines and no solution. On the other hand, if you have all of the above except that then you have two representations of the same line and therefore an infinite number of solutions. John

Triangles/310449: The angles of the triangle are 30°-30°-120°. Two of the sides lengths are 2 times the square root of 3. What is the length of the third side? 1 solutions Answer 221960 by solver91311(16872)   on 2010-06-01 18:49:36 (Show Source): You can put this solution on YOUR website! The hypotenuse is the longest side of a right triangle. It is always the side opposite the right angle. Since your 30-30-120 triangle doesn't have a right angle, it is not a right triangle, therefore it does not have a hypotenuse. Having said that, if you want to know the measure of the longest side of your triangle, even though it isn't called a hypotenuse, use the Law of Cosines. Let c represent the measure of the longest side, and let a and b represent the measures of the other two equal length sides. Let C represent the measure of the largest angle. Then which, for our purposes will be more conveniently written: Plugging in the numbers given: You can do your own arithmetic. Hint: and John

Systems-of-equations/310450: What is the solution to p-4=-9+p 1 solutions Answer 221956 by solver91311(16872)   on 2010-06-01 18:35:53 (Show Source): You can put this solution on YOUR website! Add to both sides of your equation. This results in the absurdity that . Therefore, the solution set for this equation is the empty set. John

Rational-functions/310403: Find the domain of the function: 1 solutions Answer 221940 by solver91311(16872)   on 2010-06-01 16:55:16 (Show Source): You can put this solution on YOUR website! The domain of a rational function is all real numbers EXCEPT any number that would cause the value of the denominator to be zero. Take the denominator and set it equal to zero: and then solve for Factor: Therefore, , , and must be excluded from the domain. Using interval notation, the domain is: Or in set-builder notation: John

Travel_Word_Problems/310408: Which country technology wise , reduce its travel distance by road from 3 hours to traveling by sea to 45mins? 1 solutions Answer 221938 by solver91311(16872)   on 2010-06-01 16:42:14 (Show Source): You can put this solution on YOUR website! What????????

Rational-functions/310411: Find the zeros of the function by factoring or algebraically: please note the x^3 was meant to be atop the fraction with the 36x so it should be x^3-36x atop the fraction. 1 solutions Answer 221935 by solver91311(16872)   on 2010-06-01 16:41:11 (Show Source): You can put this solution on YOUR website! Factor the numerator: Then set the function equal to zero and multiply by the denominator: Hence, or or Note: I put parentheses around your numerator so that the equation editor what parts go together. John

Complex_Numbers/310370: Solve x +6(square root of x) -7 =0 1 solutions Answer 221933 by solver91311(16872)   on 2010-06-01 16:34:08 (Show Source): You can put this solution on YOUR website! Let represent then: Factor: Hence, or And then But since by definition -- the negative square root being denoted -- discard this root. So John

Volume/310407: how do i find the volume of a sphere with a radius of 7? emaples? 1 solutions Answer 221931 by solver91311(16872)   on 2010-06-01 16:24:39 (Show Source): You can put this solution on YOUR website! The volume of a sphere with radius units is given by: John

Quadratic_Equations/310344: I am trying to solve the following problem: height of a kicked ball is modeled using The question I need to solve is this: If the nearest person is 5 ft horizontally from the point of impact how high must that person reach in order to block the punt? 1 solutions Answer 221930 by solver91311(16872)   on 2010-06-01 16:21:57 (Show Source): You can put this solution on YOUR website! The only way this makes any sense whatsoever is if the height is expressed as a function of the horizontal distance () of the ball from the kicker. Hence, you want to know the value of When which is to say: You can do your own arithmetic. John

Polynomials-and-rational-expressions/310377: Hi,I would like help with this equation,/ 1 solutions Answer 221929 by solver91311(16872)   on 2010-06-01 16:13:30 (Show Source): You can put this solution on YOUR website! What equation? An equation has an equals sign in it. That is why it is called an equation. Since there is no equals sign anywhere to be seen, you don't have an equation. What is it you would like to do with this clever little rational expression? John

Triangles/310390: if a/cosb=b/cosa then prove that triangle abc is right angled triangle 1 solutions Answer 221924 by solver91311(16872)   on 2010-06-01 15:41:23 (Show Source): You can put this solution on YOUR website! Your notation is VERY confusing. Lower case letters, generally speaking, refer to the measure of the sides of a triangle, while upper case letters are generally reserved for the measures of the angles. But taking the cosine function of the measure of a side of the triangle doesn't make any sense. Hence, unless you have referred to BOTH the measure of a side of the triangle and a measure of an angle with the same variable (sort of like calling everyone in the room "George" regardless of what their names really are), you must be saying that the ratio of the measure of one of the angles to the cosine of the measure of the other acute angle is equal to the ratio of the measure of the other acute angle to the cosine of the measure of the first angle -- an assertion that is false for all except an isosceles right triangle. John

Inequalities/310394: PLEASE ANSWER ALL OF THESE... I WANT TO GO OVER MY TEST THAT MY TEACHER HAD HANDED BACK TO ME TODAY.. AND I WANT TO SEE WHERE I WENT WRONG WITH ALL OF THESE tell weather the ordered pair is a solution of the inequality. 1. x+y>-9 (0,0) 3. 2x-y>4 ; (-6,-15) 5. 5x+2y less than or equal to 8 ; (-3,6) 7. 0.5x+2.5y greater than or equal to 2 ;(0,0) 9. 0.2y-0.5x>-1 ; (-4,-8) GRAPH THE INEQUALITY 11. x-y>-4 13. 4y less than or equal to 6x-2 15. 6Y+3 GREATER THAN OR EQUAL TO -18X 1 solutions Answer 221918 by solver91311(16872)   on 2010-06-01 15:28:14 (Show Source): You can put this solution on YOUR website! And we would be delighted to help you -- right after you re-post showing the answers you got and the work you did to get them. John

Exponents/310392: -3(3f^-1g^3)^-2 I was asked to simplify this expression. The answer key given to us my my teacher says the answer should be -f^2/3g^6, but I got -f^2/27g^6. I have found misprints in the answer key for other questions; is this yet another? 1 solutions Answer 221917 by solver91311(16872)   on 2010-06-01 15:26:40 (Show Source): You can put this solution on YOUR website! Nope. The answer key is correct this time. See? John

Functions/310391: find the domain g(x)=(8)/(4-3x) 1 solutions Answer 221913 by solver91311(16872)   on 2010-06-01 15:17:19 (Show Source): You can put this solution on YOUR website! The domain of a rational function is all real numbers EXCEPT any number that would cause the value of the denominator to be zero. Take the denominator and set it equal to zero: and then solve for Therefore, must be excluded from the domain. Using interval notation, the domain is: Or in set-builder notation: John

Surface-area/310349: The base of a rectangle is on the x-axis and its two upper vertices are on the parabola y=16-x^2. of all such rectangles, what are the dimensions of the one with greatest area? 1 solutions Answer 221898 by solver91311(16872)   on 2010-06-01 13:55:08 (Show Source): You can put this solution on YOUR website! Because of the fact that the parabola is symmetric to the y-axis, the rectangle must also be symmetric to the y-axis. That means that the two lower vertices are and . The measure of the base of the rectangle is therefore . The upper vertices, being points on the parabola are: and . Therefore, the measure of the height of the rectangle is simply . Therefore the area of the rectangle is: Such a function has local extremes at the points where the first derivative is zero: Discard the negative root since we need a positive measure of length We are looking for a maximum, so we want to see if the value of the second derivative is negative at the extreme point. Which it is... So, the horizontal dimension of the largest area rectangle is And the vertical dimension is: John

Triangles/309694: Regarding a triangle which has a 90 degree angle...is there a simple formula to take a given diagonal dimension (hypotenuse) and a constant vertical / horizontal dimension relationship (16:9), and determine the vertical / horizontal dimensions from a given diagonal dimension. For example, if I use the basic 16 horizontal dimension, and the basic 9 vertical dimension, the hypotenuse will be 18.357... Hence, working in reverse, what would be the formula I could use to plug in a given hypotenuse (20, 22, 23, 24, 25, 50, 100, etc. and using the non variable 16:9 ratio, come up with the vertical and horizontal dimensions? I hope I did OK with making my question clear. Thanks. 1 solutions Answer 221534 by solver91311(16872)   on 2010-05-30 01:38:19 (Show Source): You can put this solution on YOUR website! for your long dimension and for your short dimension Where is your variable screen diagonal

Number-Line/309717: How do I graph the portion of the number line containing all points for which x > 6 and x < y 1 solutions Answer 221533 by solver91311(16872)   on 2010-05-30 01:27:53 (Show Source): You can put this solution on YOUR website! That would depend a great deal on the value of y

Travel_Word_Problems/309291: your brother leaves on a trip, fogetting his suitcase. you know that he normally drives at a speed of 55 mph. you do not discover the suitcase until 1 hour later after he has left. if you follow him at a speed of 65 mph, how long will it takes you to catch up with him? 1 solutions Answer 221197 by solver91311(16872)   on 2010-05-27 23:34:48 (Show Source): You can put this solution on YOUR website! If he has been gone an hour and travels at 55 mph, then he is 55 miles ahead of you. You are going 10 mph faster. How long would it take to go 55 miles at 10 miles per hour? John

Quadratic-relations-and-conic-sections/309295: x"2/9 + y"2/49 = 1 1 solutions Answer 221196 by solver91311(16872)   on 2010-05-27 23:32:30 (Show Source): You can put this solution on YOUR website! What would you like to do with this ellipse? John

Problems-with-consecutive-odd-even-integers/309069: Set I contains six consecutive integers . set J contains all integers that result from adding 3 to each of the integers in set I and also contains all integers that result from subtracting 3 from each of the integers in set I . how many more integers are there in set J than in set I ? a 0 B 2 C 3 D 6 E 9 1 solutions Answer 221195 by solver91311(16872)   on 2010-05-27 23:30:32 (Show Source): You can put this solution on YOUR website! The elements of I are 3 less than the smallest one is Then 3 less than the next smallest is And 3 less than the next smallest is On the other end, Altogether, 6 additional elements. John

Rectangles/309283: the perimeter is 420yds the width is 3/4 of the length, what is the length and width. 1 solutions Answer 221183 by solver91311(16872)   on 2010-05-27 22:12:47 (Show Source): You can put this solution on YOUR website! Just solve for then multiply by to find John