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Quadratic_Equations/199573: Can anyone help me with this one? It requires me to determine the number of solutions and classify the type of solutions for each of the following equations? Justify the answer please? x^2-4x-77=0 1 solutions Answer 149969 by solver91311(16885)   on 2009-06-09 23:43:08 (Show Source): You can put this solution on YOUR website! Calculate the discriminant. Then evaluate: Two real and unequal roots. One real root with a multiplicity of two. That is to say that the trinomial is a perfect square and has two identical factors. A conjugate pair of complex roots of the form where is the imaginary number defined by John

Linear-systems/199464: This question is from textbook intermediate algebra The question is asking to solve each system by any method, if possible. If a system is inconsistent or if the equations are dependant, state this. Can you please help me solve # 39 in study set 3.2 under try it yourself. 1 solutions Answer 149929 by solver91311(16885)   on 2009-06-09 18:24:27 (Show Source): You can put this solution on YOUR website! We don't have your book. Repost and submit the problem. John

Geometry_Word_Problems/199523: Canadian Postal Service regulations require that the sum of three dimensions of a rectangular package not exceed 3 m. What are the dimensions of the largest rectangular box with square ends that can be mailed? 1 solutions Answer 149918 by solver91311(16885)   on 2009-06-09 17:23:01 (Show Source): You can put this solution on YOUR website! I have to presume that you are a calculus student and are familiar with the process of finding the first and second derivatives of a polynomial function because that is the only way I know to solve this problem. Another assumption that I have to make is that by "largest rectangular box" you mean the "rectangular box with the largest volume". Given all of that: The restriction is that the sum of the dimensions be not exceed 3 m and that at least one pair of opposite faces of the rectangular box are squares. Let represent the measure of the side of one of the square faces. Then we can say that the remaining dimension is . The volume is the product of the three dimensions so the volume function with respect to the dimension of the square end is: The feasible domain for in this situation is because if then you obviously have a zero volume box and if then the length must be zero because 2 times 1.5 is 3, and again you have a zero volume box. We are interested in finding a local maximum of the volume function on the interval (0,1.5). Take the first derivative: Set the first derivative equal to zero and solve (Fermat's Theorem) Hence which must be excluded because it is not in the feasible interval, or Now that we know that there is a critical point at we need to apply the second derivative test to determine if it is a maximum, minimum, or a possible inflection point. Take the second derivative: Therefore V(1) is a local maximum. That means that the measure of each side of the square end of the box must be 1, and since 3 - 2 = 1, the measure of the length must be 1 as well. The maximum volume rectangular box for a given sum of dimensions is a cube. John

Geometry_Word_Problems/199522: Ship A is traveling due east at 18 km/h as it passes a point 40 km due south of Ship B, which is traveling due south at 16 km/h. How much later are the ships nearest each other? 1 solutions Answer 149914 by solver91311(16885)   on 2009-06-09 16:44:26 (Show Source): You can put this solution on YOUR website! The time it takes Ship B to travel 40 km at 16 km/hr. John

Probability-and-statistics/199498: i've tried answering this, but i don't know where to start. The probablity that a door-to door salesman convinces a customer to buy is 0.65. Assume that the sales are independent find the probability that the salesman makes a sale before reaching the fifth house. 1 solutions Answer 149913 by solver91311(16885)   on 2009-06-09 16:42:23 (Show Source): You can put this solution on YOUR website! This is one where it is much simpler to calculate the probability of the event that is opposite to the desired event. First of all, "Before the fifth house..." means that he will make a sale while visiting houses number 1 through 4. If he has a 0.65 probability of making a sale, then there is a 1 - 0.65 = .35 chance that he will NOT make a sale at any given house. Making a sale at at least one of the first four houses is 1 minus the probability that he makes no sales at all at the first four visits, so: Therefore the probability that you are looking for is: You get to do your own arithmetic. John

Divisibility_and_Prime_Numbers/199515: the following scores were recorded on a 100-point examination 95,75,76,86,96,71,68,81,95,76,69,82,93,88,94 find the mean and median final examination scores? 1 solutions Answer 149909 by solver91311(16885)   on 2009-06-09 16:17:58 (Show Source): You can put this solution on YOUR website! Mean: Add up all of the data elements, then divide the sum by the number of data elements. Median: Put the scores in numerical order. Since there are an odd number of them, the median is the one in the middle. If there had been an even number of data elements, then the median would be the mean of the two in the middle. John

Divisibility_and_Prime_Numbers/199516: A) find the mode of the following set of numbers B) Find the range of the following set of numbers 11,13,13,12,11,12,13,10,14 1 solutions Answer 149908 by solver91311(16885)   on 2009-06-09 16:13:48 (Show Source): You can put this solution on YOUR website! A. Count the number of times that each data element appears in the list. The one that appears (or the ones that appear) most frequently is the mode. B. Find the largest data element, then find the smallest. Subtract the smallest from the largest. The difference is the range. John

Quadratic_Equations/199506: 1) what type of solution do you get for a quadratic equation where D<0? give the reasons for your answer. Also provide an example of such a quadratic equation and find the solution of the equation. 2) create a real-life situation that fits into the equation(x+3)(x-5)=0 and express the situation as the same equation. 1 solutions Answer 149907 by solver91311(16885)   on 2009-06-09 16:10:23 (Show Source): You can put this solution on YOUR website! Two real and unequal roots. One real root with a multiplicity of two. That is to say that the trinomial is a perfect square and has two identical factors. A conjugate pair of complex roots of the form where is the imaginary number defined by Say you want to build a box to store your note cards. All you know is that your note cards are 2 inches longer than they are wide and the area of the cards is 15 square inches. What is the minimum dimensions for the inside of the box if you want to store no more than a 4 inch tall stack of note cards? Let represent the length. Then is the width. The area is the length times the width, so: John

Polynomials-and-rational-expressions/199507: How do you solve 10/x+2 - 3x+3/x 1 solutions Answer 149904 by solver91311(16885)   on 2009-06-09 15:55:56 (Show Source): You can put this solution on YOUR website! You don't. There is nothing to solve. Since there is no equals sign, there is no equation. You can simplify the expression, but that is not what you asked. John

Numeric_Fractions/199509: This question is from textbook Intermediate Algebra There is a circle divided into 3 parts, one part is 3/4, one part is 1/3 and we are asked to find what the remaining fractional part would be. I know how to add and subtract fractions, but I am unclear as to add or subtract these two numbers to get my answer. 1 solutions Answer 149903 by solver91311(16885)   on 2009-06-09 15:50:28 (Show Source): You can put this solution on YOUR website! And you are unclear with very good reason. If you add and you get a sum greater than one, namely , so there is less than nothing left for the alleged third piece of your circle. Sort of the way our Government does arithmetic, except they use bigger numbers and it is real money. John

Square-cubic-other-roots/199483: hey ya all thanx to your help I havent be around here for a while...but even the best of luck faids away sometimes, and I am back again, with problems again, would someone be as nice as to help me out here again? =) Please My problem; Approximate to the nearest tenth, the real root of the equation f(x)=x^3-4=0, I dont at all get this one. 1 solutions Answer 149892 by solver91311(16885)   on 2009-06-09 13:11:45 (Show Source): You can put this solution on YOUR website! Finding the real root of this cubic equation is pretty straightforward. Take the cube root of both sides: Use your calculator and round to the nearest tenth. But I suspect that you are bothered by the fact that the question specifies the real root which implies that there is something other than the real root available. Indeed there is. Since this is a 3rd degree polynomial, the Fundamental Theorem of Algebra guarantees that there are exactly 3 factors of the form . Note that can be considered the difference of two cubes to the extent that 4 is the cube of the cube root of 4. The factorization of the difference of two cubes is: Which, in the case of your problem would look like: The first factor yields the same real root that we calculated earlier. The second factor is a quadratic that is guaranteed to have two factors, but if you use the discriminant, you will see that the roots of that quadratic are a conjugate pair of complex numbers. Hence, the original given equation has one real and two complex roots. John

Rectangles/199484: Problems like these always make sure I come back to this website for help =( ; Find the dimension of a rectangle “a” with the greatest area whose perimeter is 30 ft. 1 solutions Answer 149891 by solver91311(16885)   on 2009-06-09 12:41:42 (Show Source): You can put this solution on YOUR website! The perimeter of a rectangle is given by the formula: Where is the measure of the length of the rectangle and is the measure of the width. Solving for : The area of a rectangle is given by the formula: Substituting the expression for derived earlier, you can write a function for the area of the rectangle in terms of the width where the perimeter is a constant: This is a quadratic function whose graph is a parabola. Since the lead coefficient is <0, the parabola opens downward and the vertex of the parabola represents a maximum. Since the independent variable is the width and the value of the function is the area, the coordinates of the vertex will tell us the width in terms of the perimeter that gives us the maximum area, and the value of that maximum area, again in terms of the perimeter. A parabola represented by has a vertex at the point: For the area function derived above, and , so: Hence, the maximum area is obtained when , which means that and therefore also. Therefore, the maximum area for a given perimeter is obtained when the rectangle is actually a square with side measure one-fourth of the perimeter. John

real-numbers/199490: This question is from textbook algebra find the root for these radicals tell whether they are rational or irrational 1 solutions Answer 149890 by solver91311(16885)   on 2009-06-09 12:14:29 (Show Source): You can put this solution on YOUR website! If you think because you were asked for the identifying information for your textbook that we have access to your book, you are mistaken. In general, we don't have your textbook available. Think about it for a minute: 1000+ mathematics textbooks out there at $50 to $250 bucks each, and we are all unpaid volunteers. The textbook identification information is so that the website can improve the ability of students to find problems that have been previously solved. You have to post the problem with which you are having difficulty. One problem per post, please. And show your work so far. John

Numeric_Fractions/199485: I tried this problem by myself but not sure if I got it right, and have the feeling that I dont; If x varies as y and x=2 when y=8, find x when y=17; so y=k/x 8=k/2 k=16 y=16/x y=17 so 17=16/x x=16/17 I know I am a mess when it comes to algebra..... 1 solutions Answer 149889 by solver91311(16885)   on 2009-06-09 12:05:53 (Show Source): You can put this solution on YOUR website! Slight mis-step there. You wrote: . That means varies inversely as . If varies as (meaning directly as), then you want: So if when , then So: On the other hand, if you simply left out the word 'inversely' when you posted the problem (and I think that may be the case because I have seen this particular problem before) then you did it correctly. John

Numeric_Fractions/199482: I guess algebra just really isnt my strongest point, could someone help me here, I totally didnt understand this one; If y varies directly as x and inversely as z^2, and y= 3 when x=2 and z=4, what is the value of x when y=9 and z=4? 1 solutions Answer 149888 by solver91311(16885)   on 2009-06-09 11:55:49 (Show Source): You can put this solution on YOUR website! If varies directly as and inversely as , then you can say: So if when and , then you can say: Solve for : Now you can say: And if you want to know the value of when and , then just make the substitutions: And solve for . John

Money_Word_Problems/199430: Suppose you are an event coordinator for a large performance theater. One of the hottest new Broadway musicals has started to tour, and your city is the first stop on the tour. You need to supply information about projected ticket sales to the box office manager. The box office manager uses this information to anticipate staffing needs until the tickets sell out. You provide the manager with a quadratic equation that models the expected number of ticket sales for each day x. ( is the day tickets go on sale). a. Does the graph of this equation open up or down? How did you determine this? b. Describe what happens to the tickets sales as time passes? c. Use the quadratic equation to determine the last day that tickets will be sold. (Note: Write your answer in terms of the number of days after ticket sales begin.) d. Will tickets peak or be at a low during the middle of the sale? How do you know? e. After how many days will the peak or low occur? f. How many tickets will be sold on the day when the peak or low occurs? g. What is the point of the vertex? How does this number relate to your answers in parts e and f? h. How many solutions are there to the equation ? How do you know? i. What do the solutions represent? Is there a solution that does not make sense? If so, in what ways does the solution not make sense? 1 solutions Answer 149858 by solver91311(16885)   on 2009-06-08 22:16:31 (Show Source): You can put this solution on YOUR website! Can't help you unless you share the actual function, except in very general terms. If the coefficient on the term is positive, then the curve opens upward, otherwise it opens down. From the way part c is worded, I suspect it opens down -- otherwise, there wouldn't be a "last day" With that presumption, the ticket sales will increase to a peak and then fall off (part b) To answer part c, set the function equal to zero and then solve the quadratic. You would exclude any zero or negative root because the last day can't be the first day and the last day can't happen before the first day. part d: see the answer to part b. Given a quadratic function , the vertex occurs at the x-coordinate . That will answer part e. Evaluate the function at the answer to part e, that is substitute that value for x in the function and calculate the value of the function to get part f. The answers to e and f are the coordinates of the ordered pair representing the vertex. for part h, I can't exactly say without looking at the actual function, however using the assumption I used for part c, the vertex is probably above the x-axis, hence the curve intersects the x-axis in two places -- hence two solutions. Each solution represents a day when ticket sales are zero. A negative solution would not make sense in terms of what is going on. Of course ticket sales are zero at any time before you start selling them -- but knowing that isn't very helpful when trying to model the situation. John

Triangles/199428: How Far away from a building are you standing if you know the building is 295' high and the angle of elevation to the top of the building is 65 degrees? 1 solutions Answer 149856 by solver91311(16885)   on 2009-06-08 21:44:01 (Show Source): You can put this solution on YOUR website! I just did this one. Question 199428 John

Triangles/199425: this is my math problem. How Far away from a building are you standing if you know the building is 295' high and the angle of elevation to the top of the building is 65 degrees? and i dont know how to find the answer. 1 solutions Answer 149855 by solver91311(16885)   on 2009-06-08 21:38:53 (Show Source): You can put this solution on YOUR website! The cotangent of an angle is equal to the measure of the adjacent side divided by the measure of the opposite side. So: Make sure your calculator is in Degrees mode. By the way, since the measure of the height of the building is given to whole number precision, your answer should be expressed to no greater precision, that is, it should be rounded to the nearest foot. John

Quadratic_Equations/199380: Analytic Geometry 11. Determine the shortest distance from the origin to the line represented by y=1/2x-2. Thank you!@@ 1 solutions Answer 149854 by solver91311(16885)   on 2009-06-08 21:31:10 (Show Source): You can put this solution on YOUR website! The given line is already in slope-intercept form, so you can determine the slope by inspection. Determine the slope of a line perpendicular to the given line by calculating the negative reciprocal of the slope of the given line. Use the point-slope form of the equation of a line to determine the equation of a line perpendicular to the given line that passes through the origin. Where is the negative reciprocal of the slope of the given line and is the origin, (0,0). The given line and its perpendicular through the origin form a system of equations. Solve the system for the point of intersection. Finally use the distance formula: Where and are the point of intersection and the origin to calculate the distance from the origin to the given line. John

Functions/199422: Hello! If a ball is thrown directly upward with a velocity of 31 ft/s, its height (in feet) after t seconds is given by y = 31t - 16t^2. What is the maximum height attained by the ball? thanks! 1 solutions Answer 149853 by solver91311(16885)   on 2009-06-08 21:13:59 (Show Source): You can put this solution on YOUR website! Using the model you provided (which, by the way, is a very poor model for a ball being thrown straight upward at such a low initial velocity -- more to follow), you first need to set your function into standard form, that is . Notice that this is the equation of a parabola. The lead coefficient is less than zero, so it opens downward. Therefore, the vertex is a maximum value for the function. The independent variable coordinate of the vertex of any parabola of the form is given by , hence: , and the value of the function at that time value is: You get to do your own arithmetic. The reason that your model for this particular situation is such a poor one is that it assumes that the initial height is zero. That means that you are either modeling this based on zero height being the height above the ground at which the thrower's hand was when the ball was released (a very odd choice for a height baseline indeed) or somehow this person was able to release the ball straight upward at 31 feet per second while their hand was touching the ground. If you look at the numbers here, the time at max height is just a little less than 1 second, meaning that the height calculated by the model is going to be very nearly 15 feet. But a person in the height range of 5 to 6 feet tall is going to release a ball thrown straight up at about 6 to 7 feet above the ground. That changes the actual height reached by this particular thrown ball by something on the order of 40%. The correct height model, neglecting atmospheric friction effects, is: Where or depending on the system of units and assuming you are on planet Earth, is the initial velocity, and is the initial height. John

Linear_Equations_And_Systems_Word_Problems/199414: I need to solve for the discriminate and determine the number of solution and classify it for the equation: square root of 2x^2-4x-7sq root of 2-0 can you help me? 1 solutions Answer 149852 by solver91311(16885)   on 2009-06-08 21:11:30 (Show Source): You can put this solution on YOUR website! I don't know what a 'discriminate' means with respect to a quadratic polynomial. And you do not have an equation since there is no equals sign. So I can't precisely answer the question you posed. However, if you meant: "I need to calculate the discriminant, determine the number of solutions, and classify the solutions for the equation: sqrt(2)x^2-4x-7*sqrt(2)=0" Then I can help. Given a quadratic equation in standard form, namely: You can calculate the discriminant, , by substituting the given coefficients into: Then you can classify the roots of the equation thus: Two real and unequal roots. One real root with a multiplicity of two. That is to say that the trinomial is a perfect square and has two identical factors. A conjugate pair of complex roots of the form where is the imaginary number defined by For your problem: , , and , so You can do your own arithmetic and then evaluate according to the criteria listed. John

Geometric_formulas/199387: This question is from textbook geometry can you please help me solve (((Find the area of a rectangular solid with length 10, width6, and height 5. ))) 1 solutions Answer 149849 by solver91311(16885)   on 2009-06-08 20:51:51 (Show Source): You can put this solution on YOUR website! Two faces that are 10 X 6, two faces that are 10 X 5, and two faces that are 5 X 6. Compute the areas of the 6 faces and add 'em up. John

Linear_Equations_And_Systems_Word_Problems/199415: I need to find a equation in which -3 and +4 are solutions. I am really stumped 1 solutions Answer 149848 by solver91311(16885)   on 2009-06-08 20:48:41 (Show Source): You can put this solution on YOUR website! You are just doing the problem backwards. In the first place, since there are two solutions, you know that you must have a second degree equation. If you were solving a quadratic, something that looks like: You would factor the trinomial and have something that looks like: Then you would use the Zero Product Rule to say: so or so Therefore, start at the end: so Likewise: Now we have our two factors: So all you have to do is multiply the binomials using FOIL and you are done. John

expressions/199412: the quotient of 5 and a nonzero number. 1 solutions Answer 149836 by solver91311(16885)   on 2009-06-08 18:35:44 (Show Source): You can put this solution on YOUR website! John

Word_Problems_With_Coins/199411: The Hudson River flows at a rate of 5 miles per hour. A patrol boat travels 40 miles upriver, and returns in a total time of 6 hours. What is the speed of the boat in still water? 1 solutions Answer 149835 by solver91311(16885)   on 2009-06-08 18:34:05 (Show Source): You can put this solution on YOUR website! I've already solved one of these today. See problem 199336 http://www.algebra.com/algebra/homework/word/travel/Travel_Word_Problems.faq.question.199336.html Change the numbers to suit, but the process is precisely the same. John

Quadratic_Equations/199383: Analytic Geometry 13. a) Draw the triangle with vertices P(-2,-2),Q(2,4), and R(8,0). b) Show algebraically that PQR is a right triangle. c)Is PQR also an isosceles triangle? Use algebraic reasoning to justify your anser. I need help in this question.....PLEASE... Thank you very much!!!!!! 1 solutions Answer 149834 by solver91311(16885)   on 2009-06-08 18:21:03 (Show Source): You can put this solution on YOUR website! Use the formula for the slope of a line passing through two points, namely: Where and are the two points. Do this twice: once for segment PQ and once for segment QR. Since: If then and PQR is a right triangle by definition. The triangle is isosceles if the measure of PQ is equal to the measure of QR. Use the distance formula, where and are the coordinates of the endpoints of the segment to calculate the measures of PQ and QR. If they are equal, the triangle is isosceles; otherwise not. John

Graphs/199402: Find the equation, in standard form, of the line perpendicular to 2x + 3y = -5 and passing through (3, -5). Write the equation in standard form, with all integer coefficients. 1 solutions Answer 149832 by solver91311(16885)   on 2009-06-08 18:05:00 (Show Source): You can put this solution on YOUR website! First, calculate the slope of the given line by putting the equation into slope-intercept form, namely: Once you have completed that step, the coefficient on the term will be the slope. Next, calculate the slope of a line perpendicular to the given line using: In other words, take the negative reciprocal of the slope of the given line. Using the calculated slope for the desired line and the given point, use the point-slope form of the equation of a line to derive an equation representing the desired line. Where is the slope calculated above and are the coefficients of the given point. Putting it into standard form with integer coefficients is an exercise for the student. Standard form is: John

Geometric_formulas/199389: This question is from textbook geometry can you please help me solve ((( Find the total area of a cube with edge 4. ))) 1 solutions Answer 149830 by solver91311(16885)   on 2009-06-08 17:58:26 (Show Source): You can put this solution on YOUR website! A cube has six identical faces. Each face is a square with area , so the total surface area of a cube is six times the area of one face, or: John

Travel_Word_Problems/199398: IF A CAR IS TRAVELING AT 30 MPH, HOW LONG WILL IT TAKE TO DRIVE 1 MILE? 1 solutions Answer 149829 by solver91311(16885)   on 2009-06-08 17:55:17 (Show Source): You can put this solution on YOUR website! All CAPS is the electronic equivalent of shouting and is therefore both rude and annoying. An object traveling at miles per hour is traveling at hours per mile. John

decimal-numbers/199399: i need help expressing 5E-5 in decimal form when given as an answer on a calculator 1 solutions Answer 149828 by solver91311(16885)   on 2009-06-08 17:51:20 (Show Source): You can put this solution on YOUR website! The leading number(s), 5 in your case, are the significant digits. Write those down. Then move the decimal point the number of places indicated by the number following the E. If it is negative, move the decimal point to the left; positive to the right. Another example: John

Reduction-of-unit-multipliers/199397: This question is from textbook Algebra 2 with trigonometry xcubed minus xsquard+x-1=0 find the complex zero 1 solutions Answer 149827 by solver91311(16885)   on 2009-06-08 17:34:40 (Show Source): You can put this solution on YOUR website! You can never find the complex zero of a polynomial function of any degree. That is because complex zeros always come in pairs, namely conjugate pairs of the form . Having said that, let's find all three of the zeros of this cubic. All cubics with rational coefficients must have at least one real zero, and it must be rational. That is because irrational zeros of polynomial equations with rational coefficients also always come in pairs. Using the Rational Zero Theorem: If a polynomial function, written in descending order, has integer coefficients, then any rational zero must be of the form , where is a factor of the constant term and is a factor of the leading coefficient. Examining the lead and constant coefficients of the given polynomial, we can see that the only possible rational zeros are A bit of polynomial long division results in: (verification of this step is left as an exercise for the student) Therefore, is a real root of and the other two roots are also roots of So the two complex roots are: and John