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 Equations/165615: This question is from textbook Amscos Integrated Algebra 1 1/2t-6+8(t-9+7)=4t*3-5+2+3/4t1 solutions Answer 122082 by jim_thompson5910(28717)   on 2008-11-03 19:08:39 (Show Source): You can put this solution on YOUR website! Start with the given equation Multiply Distribute. Multiply both sides by the LCD to clear any fractions. Distribute and multiply. Combine like terms on the left side. Combine like terms on the right side. Add to 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 answer is which approximates to .
 Linear-systems/165439: Solve the system by addition or substitution. -9x - 3y = 22 y = -3x - 61 solutions Answer 121954 by jim_thompson5910(28717)   on 2008-11-03 00:12:53 (Show Source): You can put this solution on YOUR website! Start with the given system Plug in into the first equation. In other words, replace each with . Notice we've eliminated the variables. So we now have a simple equation with one unknown. Distribute Combine like terms on the left side Subtract 18 from both sides Combine like terms on the right side Since this equation is never true for any x value, this means there are no solutions.
 Linear-systems/165449: Solve the system by graphing. x + y = 3 x + y = -11 solutions Answer 121953 by jim_thompson5910(28717)   on 2008-11-03 00:09:23 (Show Source): You can put this solution on YOUR website! Start with the given system of equations: In order to graph these equations, we must solve for y first. Let's graph the first equation: Start with the first equation. Subtract from both sides. Rearrange the terms and simplify. Now let's graph the equation: Graph of . ------------------------------------------------------------------- Now let's graph the second equation: Start with the second equation. Subtract from both sides. Rearrange the terms and simplify. Now let's graph the equation: Graph of . ------------------------------------------------------------------- Now let's graph the two equations together: Graph of (red). Graph of (green) From the graph, we can see that the two lines are parallel, which means that they will never intersect. So there are no solutions. This means that the system of equations is inconsistent.
 logarithm/165416: This is not from a textbook, just from a worksheet my teacher made up. The question is : 4logx = 4 I think that you can move the first 4 to make the question logx^4=4 , but that is as far as I have gotten. Thank You For Your Help!1 solutions Answer 121913 by jim_thompson5910(28717)   on 2008-11-02 19:36:31 (Show Source): You can put this solution on YOUR website!Note: I'm assuming that the base is 10 Start with the given equation Divide both sides by 4 Divide and simplify Rewrite the equation using the property: ====> Raise 10 to the first power to get 10 So the solution is
 Rational-functions/165403: Find all horizontal/vertical asymptotes and all x-intercepts: 1 solutions Answer 121908 by jim_thompson5910(28717)   on 2008-11-02 19:33:28 (Show Source): You can put this solution on YOUR website!x-intercepts: Start with the given equation Plug in y=0 Set the numerator equal to zero Factor the left side (note: if you need help with factoring, check out this solver) Now set each factor equal to zero: or or Now solve for x in each case So our answers are or So the x-intercepts are (3,0) and (2,0) =========================================================== Asymptotes: Looking at the numerator , we can see that the degree is since the highest exponent of the numerator is . For the denominator , we can see that the degree is since the highest exponent of the denominator is . Oblique Asymptote: Since the degree of the numerator (which is ) is greater than the degree of the denominator (which is ), there is no horizontal asymptote. In this case, there's an oblique asymptote. To find the oblique asymptote, simply use polynomial long division to find it. So the quotient is , this means that the oblique asymptote is -------------------------------------------------- Vertical Asymptote: To find the vertical asymptote, just set the denominator equal to zero and solve for x Set the denominator equal to zero Add 1 to both sides Combine like terms on the right side So the vertical asymptote is Notice if we graph , we can visually verify our answers: Graph of with the x-intercepts (3,0) and (2,0), the oblique asymptote (blue line) and the vertical asymptote (green line)
 Linear-systems/165385: This question is from textbook Algebra 2 How do you use substitution to solve each system of equations. {12x+y=21 {18x-3y=-361 solutions Answer 121891 by jim_thompson5910(28717)   on 2008-11-02 17:42:25 (Show Source): You can put this solution on YOUR website!Start with the given system of equations: 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 Add 63 to both sides Combine like terms on the right side Divide both sides by 54 to isolate x Reduce -----------------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 and reduce. (note: if you need help with fractions, check out this solver) -----------------Second Answer------------------------------ So the second part of our answer is: -----------------Summary------------------------------ So our answers are: and which form the ordered pair
 Functions/165387: This is my problem and I believe I have the answer (C(x))=3x+27,000 but I don't know the proper process for illustrating this. Please help! # If the relationship between the total cost and the number of units produced is linear and if the total cost increases by $3 for each additional unit made: 1. Find the equation of the total cost function C(x) using the fact that C(1,000) = 30,000 (dollars) where x is the number of units produced. 1 solutions Answer 121890 by jim_thompson5910(28717) on 2008-11-02 17:39:43 (Show Source): You can put this solution on YOUR website!Since the "total cost increases by$3 for each additional unit made", this means that the slope is . Also, because C(1,000) = 30,000 , this means that we have the point (1000,30000). So let's find the equation of the line that has a slope of 3 and goes through the point (1000,30000) If you want to find the equation of line with a given a slope of which goes through the point (1000,30000), you can simply use the point-slope formula to find the equation: ---Point-Slope Formula--- where is the slope, and is the given point So lets use the Point-Slope Formula to find the equation of the line Plug in , , and (these values are given) Distribute Multiply and to get Add 30000 to both sides to isolate y Combine like terms and to get ------------------------------------------------------------------------------------------------------------ Answer: So the equation of the line with a slope of which goes through the point (1000,30000) is: which is now in form where the slope is and the y-intercept is So in function notation, the cost function is . So you are correct.
 Polynomials-and-rational-expressions/165290: f(x)=4x^3-3x^2+2x-1. Find f(1+2i). Please help me out, I really don't have any idea on what to do!!!1 solutions Answer 121859 by jim_thompson5910(28717)   on 2008-11-02 12:14:27 (Show Source): You can put this solution on YOUR website!Note: it's helpful to remember that Start with the given function Plug in . In other words, replace each "x" with "1+2i" Break up to get FOIL to get FOIL to get Distribute Combine like terms. So the answer is
 Quadratic_Equations/165304: given f(x)= find all zeros (real and imaginary)1 solutions Answer 121857 by jim_thompson5910(28717)   on 2008-11-02 12:09:42 (Show Source): You can put this solution on YOUR website! Start with the given function Plug in Factor to get . Note: use the difference of cubes formula. or or Set each individual factor equal to zero Solve the first two equations to get or . These are the first two zeros Now let's solve Notice we have a quadratic equation in the form of where , , and Let's use the quadratic formula to solve for x Start with the quadratic formula Plug in , , and Square to get . Multiply to get Subtract from to get Multiply and to get . Simplify the square root (note: If you need help with simplifying square roots, check out this solver) or Break up the expression. So the zeros of are or which approximate to or ======================================================= Answer: So in total, the 4 zeros are: , , or Which in decimal form are , , or
 Complex_Numbers/165291: f(x)=4x^3-3x^2+2x-1. Find f(1+2i). Please help me out, I really don't have any idea on what to do!!!1 solutions Answer 121854 by jim_thompson5910(28717)   on 2008-11-02 11:54:34 (Show Source): You can put this solution on YOUR website!Note: it's helpful to remember that Start with the given function Plug in . In other words, replace each "x" with "1+2i" Break up to get FOIL to get FOIL to get Distribute Combine like terms. So the answer is
 Linear-systems/165293: Solve by the substition method 8m+n=7 m-6n=561 solutions Answer 121853 by jim_thompson5910(28717)   on 2008-11-02 11:42:58 (Show Source): You can put this solution on YOUR website! Start with the given system of equations: Now in order to solve this system by using substitution, we need to solve (or isolate) one variable. I'm going to solve for n. So let's isolate n 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 Add 42 to both sides Combine like terms on the right side Divide both sides by 49 to isolate m 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 point
 Linear-systems/165294: Solve the system of equations using the substitution method. If the answer is a unique solution, present it as an ordered pair: (x, y). If not, specify whether the answer is “no solution” or “infinitely many solutions.” 3x + y = 7 4x – y = 21 1 solutions Answer 121851 by jim_thompson5910(28717)   on 2008-11-02 11:39:30 (Show Source): You can put this solution on YOUR website! Start with the given system of equations: 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 the negative Combine like terms on the left side Add 7 to 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 point Now let's graph the two equations (if you need help with graphing, check out this solver) From the graph, we can see that the two equations intersect at . This visually verifies our answer. graph of (red) and (green) and the intersection of the lines (blue circle).
 Triangles/165224: This question is from textbook Lets Review Math A QUESTION: In isosceles triangle ABC the ratio of the measure of vertex angle A to the measure of angle B is 2 to 5. Find angle C. 1st APPROACH: we know that in an isosceles triangle two angles are the same and the sides opposite them are also the same. Since this problem doesn't tell you which is the base vertex in this isosceles triangle, I have assumed it to be 5x. Because of "RULE:" the length of each side is less than the sum of the lengths of the other two sides and greater than the difference between these lengths....3less than x less than 7. Therefore x is in the range of 4, 5, 6. 2x+5x+5x=180 12x=180 x=15 Angle C is 75 degrees 2nd APPROACH: we know that in an isosceles triangle two angles are the same and the sides opposite them are also the same. Since we already used 5x as the base vertex in our previous approach lets assume 2x is the base angle, this time, because it's an isosceles triangle. 2x+2x+5x=180 9x=180 x=20 Angle C is 40 degrees BOOKS ANSWER IS 100 degrees, why? 1 solutions Answer 121825 by jim_thompson5910(28717)   on 2008-11-02 00:51:12 (Show Source): You can put this solution on YOUR website!Here's a quick note about what the ratio is really saying: Since the ratio is vertex angle A to the measure of angle B, this means that we have this ratio: A:B ---> 2:5 So this means that the vertex angle is SMALLER than the base angle. However, if you have a base angle of 40 degrees, this means that the vertex angle is which means that the vertex angle is larger (not smaller) than the base angle. So this effectively rules out the "2nd Approach" answer. Remember, the ratio is vertex:base angle (not the other way around) However, I agree with your first approach. That answer is correct. You'll find that the base angles are 75 degrees and the vertex angle is 30 degrees. Since the ratio of 30:75 reduces to 2:5, this verifies the answer. Note: it is IMPOSSIBLE to have both an isosceles triangle AND have a base angle that is greater than 90 degrees. Why? Base angles are equal. So if there is an obtuse base angle, there are really 2 obtuse angles in the triangle (which is impossible; triangles can only have at most one obtuse angles). So this means that either the book is wrong or maybe you read the wrong answer.
 Polynomials-and-rational-expressions/165265: 7.) Solve by completing the square. 2x^2+x-6=0 the answer shows -2,3/2 I am not sure how they got the answer?1 solutions Answer 121823 by jim_thompson5910(28717)   on 2008-11-02 00:18:28 (Show Source): You can put this solution on YOUR website! Start with the given expression. Factor out the coefficient . This step is very important: the coefficient MUST be equal to 1. Take half of the coefficient to get . In other words, . Now square to get . In other words, Now add and subtract inside the parenthesis. Make sure to place this after the "x" term. Notice how . So the expression is not changed. Group the first three terms. Factor to get . Combine like terms. Distribute. Multiply. So after completing the square, transforms to . So . So is equivalent to . Add to both sides. Combine like terms. Divide both sides by . Reduce. Take the square root of both sides. or Break up the "plus/minus" to form two equations. or Take the square root of to get . or Subtract from both sides. or Combine like terms. -------------------------------------- Answer: So the solutions are or . Note: the order of the solutions does NOT matter.
 Linear-systems/165268: There are two numbers whose sum is 50. Three times the first is five more than twice the second. What are the numbers? Please DO answer quickly.1 solutions Answer 121822 by jim_thompson5910(28717)   on 2008-11-02 00:14:17 (Show Source): You can put this solution on YOUR website!"There are two numbers whose sum is 50" ---> "Three times the first is five more than twice the second" ---> Start with the first equation. Subtract x from both sides. Move onto the second equation Plug in Distribute. Combine like terms on the right side. Add to both sides. Combine like terms on the left side. Divide both sides by to isolate . Reduce. Go back to the previously isolated equation Plug in Subtract ---------------------------------------------------------------------- Answer: So the solutions are and which form the ordered pair (21,29)
 Subset/165241: Let c = {p,l,u,s}. How many subsets does c have?1 solutions Answer 121821 by jim_thompson5910(28717)   on 2008-11-02 00:09:52 (Show Source): You can put this solution on YOUR website!Subsets of c (16 total): {p,l,u,s}, {p,l,u},{p,l,s},{p,u,s},{l,u,s} {p,l},{p,u},{p,s},{l,u},{l,s},{u,s} {p},{l},{u},{s} {} Note: {} denotes the empty set To help confirm (not completely verify) that we have the right answer, notice how there are 16 subsets. It turns out that for any set A of "n" elements, the total number of subsets of set A is . Since there are 4 elements in set c, and , this helps confirm that we at least have the right number of subsets listed.
 Miscellaneous_Word_Problems/165217: The measures of the angles of a certain triangle are consecutive even integers. Find the measures.1 solutions Answer 121777 by jim_thompson5910(28717)   on 2008-11-01 16:13:55 (Show Source): You can put this solution on YOUR website!Let x=first angle, x+2=second angle, x+4=third angle Remember, the sum of three angles in a triangle is 180 degrees. So this means that Start with the given equation. Combine like terms on the left side. Subtract from both sides. Combine like terms on the right side. Divide both sides by to isolate . Reduce. ---------------------------------------------------------------------- Answer: So the answer is So this means that the three angles are 58, 60, and 62
 Functions/165207: state the domain of the following and provide a brief explanation for your answer: h(x)=3x^2+5x-31 solutions Answer 121776 by jim_thompson5910(28717)   on 2008-11-01 16:06:33 (Show Source): You can put this solution on YOUR website!Looking at , we can see that there are no square roots, logs, and other functions where there are restrictions on the domain. Also, we can see that the function does not have a division by x (or any combination of variables and constants). So we don't have to worry about division by zero. Since we don't have any restrictions on the domain, this shows us that the domain is all real numbers. In other words, we can plug in any number in for x So the domain of the function in set-builder notation is: In plain English, this reads: x is the set of all real numbers (In other words, x can be any number) Also, in interval notation, the domain is:
 Polynomials-and-rational-expressions/165214: Please assist me with the following problem: Determine the leading term, the leading coefficient, and the degree of the polynomial. Then classify the polynomial as constant, linear, quadratic, cubic, or quartic. f(x)=15x^2-10+0.11x^4-7x^3 1 solutions Answer 121775 by jim_thompson5910(28717)   on 2008-11-01 16:05:05 (Show Source): You can put this solution on YOUR website!First, sort the terms in descending order. So the term with the largest exponent goes first. So sorts to Leading term: The first term of the sorted terms So the leading term is ---------------------------------------- Leading Coefficient: coefficient of leading term So the leading coefficient is 0.11 ---------------------------------------- Degree of the Polynomial: This value is equal to the largest exponent of the polynomial So the degree of the polynomial is 4 since 4 is the largest exponent ---------------------------------------- Since the degree of the polynomial is 4, this classifies the polynomial as a quartic
 logarithm/165216: This question is from textbook log(base5) x - log(base5)(x-2)= log(base5) 4 Many thanks.1 solutions Answer 121774 by jim_thompson5910(28717)   on 2008-11-01 15:59:40 (Show Source): You can put this solution on YOUR website! Start with the given equation Combine the logs on the left side using the identity Since the bases of the logs on both sides are equal, this means that the arguments of the logs (the stuff inside the logs) are equal Multiply both sides by Distribute. Subtract from both sides. Combine like terms on the left side. Divide both sides by to isolate . Reduce. ---------------------------------------------------------------------- Answer: So the answer is which approximates to .
 Polynomials-and-rational-expressions/165212: Please assist me with the following problem: Using the intermediate value theorem, determine, if possible, whether the function f has a real zero between a and b f(x)=3x^2-2x-11; a=2, b=3 1 solutions Answer 121772 by jim_thompson5910(28717)   on 2008-11-01 15:37:29 (Show Source): You can put this solution on YOUR website!Let's evaluate the left endpoint a=2 Start with the given equation. Plug in . Square to get . Multiply and to get . Multiply and to get . Combine like terms. ------------------------------------------------ Let's evaluate the right endpoint b=3 Start with the given equation. Plug in . Square to get . Multiply and to get . Multiply and to get . Combine like terms. So as x changes from 2 to 3, f(x) (ie y) changes from -3 to 10 which means that the graph MUST have crossed over the x-axis somewhere in between x=2 and x=3. So this shows that there is a zero between a and b
 Inequalities/165211: This question is from textbook Holt Algebra 1 Texas Im not sure what the OR has to do with anything for example could you explain to me how to solve x-1<-1ORx-5>-1?1 solutions Answer 121768 by jim_thompson5910(28717)   on 2008-11-01 15:23:24 (Show Source): You can put this solution on YOUR website!To solve inequalities like this, simply solve each inequality individually Let's solve the first inequality : Start with the first inequality. Add to both sides. Combine like terms on the right side. --------------------------------------------------------------------- Now let's solve the second inequality : Start with the second inequality. Add to both sides. Combine like terms on the right side. So our answer is or So the solution in interval notation is: () () So the solution in set notation is: Here's the graph of the solution set Note: There is an open circle at which means that we're excluding that value from the solution set. Also, there is an open circle at which means that we're excluding that value from the solution set.
 Inequalities/165210: How exactly do I solve a compound inequality such as -31 solutions Answer 121767 by jim_thompson5910(28717)   on 2008-11-01 15:19:53 (Show Source): You can put this solution on YOUR website! Start with the given compound inequality. Subtract from ALL sides. Combine like terms on the left side. Combine like terms on the right side. So our answer is So the answer in interval notation is () So the answer in set-builder notation is Here's the graph of the solution set Graph of the solution set Note: There is an open circle at which means that we're excluding this value from the solution set Also, there is an open circle at which means that we're excluding this value from the solution set.
 Functions/165208: State the domain of the following and provide a brief explanation for you answer: m(x)=(5)/(x^2-9)1 solutions Answer 121766 by jim_thompson5910(28717)   on 2008-11-01 15:15:34 (Show Source): You can put this solution on YOUR website! Start with the given function Set the denominator equal to zero. Remember, dividing by 0 is undefined. So if we find values of x that make the denominator zero, then we must exclude them from the domain. Factor the left side (note: if you need help with factoring, check out this solver) Now set each factor equal to zero: or or Now solve for x in each case So our solutions are or Since and make the denominator equal to zero, this means we must exclude and from our domain So our domain is: which in plain English reads: x is the set of all real numbers except or So our domain looks like this in interval notation note: remember, the parenthesis excludes -3 and 3 from the domain If we wanted to graph the domain on a number line, we would get: Graph of the domain in blue and the excluded values represented by open circles Notice we have a continuous line until we get to the holes at and (which is represented by the open circles). This graphically represents our domain in which x can be any number except x cannot equal -3 or 3
 test/165192: 15.) Graph each parabola identify the vertex, intercepts, and the maximum or minimum y - value. g(x) = x^2 - 3x1 solutions Answer 121761 by jim_thompson5910(28717)   on 2008-11-01 15:05:23 (Show Source): You can put this solution on YOUR website! Graph of Vertex: X-intercepts: (0,0) and (3,0) Y-intercept: (0,0) Min value:
 Polynomials-and-rational-expressions/165203: simplify. only positive exponents (2y)^-3 over (x^5y^7)^-21 solutions Answer 121760 by jim_thompson5910(28717)   on 2008-11-01 14:54:00 (Show Source): You can put this solution on YOUR website!Let's break up the expression Numerator: Denominator: So the expression becomes which then simplifies to Now divide to get So simplifies to
 Radicals/165083: This question is from textbook Elementary and Intermediate 30.) Solve each equation and check for extraneous solutions. sqrt(a-1 -5 = 1) 1 solutions Answer 121689 by jim_thompson5910(28717)   on 2008-10-31 21:55:42 (Show Source): You can put this solution on YOUR website!I'm assuming that you mean ??? Be careful to place your parenthesis in the right spot. Start with the given equation. Add 5 to both sides. Add. Square both sides to eliminate the square root. Square 6 to get 36. Add to both sides. Combine like terms on the right side. ---------------------------------------------------------------------- Answer: So the answer is
 Evaluation_Word_Problems/165097: A circular flower bed is surrounded by a ring of grass with an Area of 314 square yards. Find the width of the ring of grass. 1 solutions Answer 121688 by jim_thompson5910(28717)   on 2008-10-31 21:52:14 (Show Source): You can put this solution on YOUR website!Are you sure that you don't want find the "radius" (instead of the "width")? Or do you want to find the diameter? Start with the area of a circle formula Plug in and replace with (this is an approximation) Divide both sides by 3.14 Divide Rearrange the equation. Take the square root of both sides (only the positive square root is considered). So the radius of the circle is about 10 yards which means that the diameter is about 20 yards (which is twice the radius)
 Polynomials-and-rational-expressions/165095: solve each equations. indicate the lower comon demominator. 4+3/x+2=4/x^2+2x 1 solutions Answer 121687 by jim_thompson5910(28717)   on 2008-10-31 21:43:32 (Show Source): You can put this solution on YOUR website! Start with the given equation Factor to get So the LCD is Multiply EVERY term by the LCD to clear the fractions. Simplify. Distribute. Subtract 4 from both sides. Combine like terms. Notice we have a quadratic equation in the form of where , , and Let's use the quadratic formula to solve for x Start with the quadratic formula Plug in , , and Square to get . Multiply to get Rewrite as Add to to get Multiply and to get . or Break up the expression. =============================================================== Answer: So the solutions are or which approximate to or
 Evaluation_Word_Problems/165101: A baseball is hit at a point 3 ft above the ground at a velocity of 100ft/s and at an angle of 45 degrees with respect of the ground. The path of the baseball is given by the function f(x) = -0.0032x2 + x +3 where f(x) is the height of the baseball (in feet) and x is the horizontal distance from home plate (in feet). What's the maximum hieght reached by the baseball?1 solutions Answer 121685 by jim_thompson5910(28717)   on 2008-10-31 21:36:09 (Show Source): You can put this solution on YOUR website!Remember, the max height occurs at the vertex. So if we find the y-coordinate of the vertex, we find the max height. Note: the same applies to the min height (if there exists a min height) In order to find the vertex, we first need to find the x-coordinate of the vertex. To find the x-coordinate of the vertex, use this formula: . Start with the given formula. From , we can see that , , and . Plug in and . Multiply. Divide So the x-coordinate of the vertex is . Now that we know the x-coordinate of the vertex, we can use it to find the y-coordinate of the vertex. Start with the given equation. Plug in . Square to get . Multiply Combine like terms. So the y-coordinate of the vertex is . So the vertex is (156.25,81.125). Since the y-coordinate is , this means that the max height is 81.125 ft.
 Evaluation_Word_Problems/165093: A 10m by 34m garden is surrounded by a walkway of uniform width (call it "x"). The total area of the garden and walkway is 640 square meters. what is the width of the walkway?1 solutions Answer 121684 by jim_thompson5910(28717)   on 2008-10-31 21:24:32 (Show Source): You can put this solution on YOUR website!Let x = width of the path With these type of problems, it helps to draw a picture. So draw two rectangles, one inside of the other, and label the inside rectangle's dimensions: Now the label the width of the path "x" (denoted in red) Since there are 2 "x" lengths per side, this means that you need to add "2x" to both the length and width of the inner rectangle to get the length and width of the outer rectangle. If this makes no sense at all, here's a visual: So the length and width of the outer rectangle is and respectively. This means that for the outer rectangle (the given area of both the walkway and the garden), and Remember, the area of any rectangle is Start with the area of a rectangle formula Plug in , and FOIL Subtract 640 from both sides. Combine like terms. Notice we have a quadratic equation in the form of where , , and Let's use the quadratic formula to solve for x. Start with the quadratic formula Plug in , , and Square to get . Multiply to get Rewrite as Add to to get Multiply and to get . Take the square root of to get . or Break up the expression. or Combine like terms. or Simplify. So the possible answers are or However, since a negative width is not possible, this means that is NOT a solution. ================================================== Answer: So the solution is which means that the width of the path is 3 meters.