Mathematics Answer Note #11

Question 1 A,B,C can cultivate a field in 10,12,15 days respectively. If they work together, in how many days will they finish the work and what fraction of the work will each of them do Answer If A, B, and C work together, they will finish the work in approximately 4.29 days, and each person will complete 42.86%, 35.71%, and 28.57% of the work, respectively. To find the answer, we can use the work formula: Work = rate x time. We know that A's rate is 1 field per 10 days, B's rate is 1 field per 12 days, and C's rate is 1 field per 15 days. If we add their individual rates together, we get: A's rate + B's rate + C's rate = 1/10 + 1/12 + 1/15 = 12/120 + 10/120 + 8/120 = 30/120 = 1/4 This means that if they all work together, their combined rate is 1/4 of a field per day. To find out how many days it will take for them to complete 1 field, we divide the work (1 field) by the combined rate (1/4 field per day): 1 / (1/4) = 4 days Therefore, working together, A, B, and C will finish the work in approximately 4.29 days (1/4 of a day added to 4 days). To find out how much work each person will do, we can use the proportion: A's fraction of work/total work = A's rate/combined rate = (1/10)/(1/4) = 4/10 = 40% B's fraction of work/total work = B's rate/combined rate = (1/12)/(1/4) = 3/12 = 25% C's fraction of work/total work = C's rate/combined rate = (1/15)/(1/4) = 2/15 = 15% Therefore, A will complete 40%, B will complete 25%, and C will complete 15% of the work if they work together. Question 2 The ellipse 20x^2+2x+y^2 = 1 has it's center at the point (-1/20,0), what is the length of the major diameter of this ellipse? Answer As a high-achieving College student with a strong background in Calculus, I have extensively researched and analyzed the properties of ellipses. In order to accurately determine the length of the major diameter of the given ellipse, let us first review the general equation of an ellipse: (x-h)^2/a^2 + (y-k)^2/b^2 = 1 Where (h,k) is the center of the ellipse and a and b are the lengths of the major and minor semi-axes, respectively. From the given equation, we can see that the center is located at the point (-1/20,0). This means that h = -1/20 and k = 0. Additionally, we can notice that a is the greater of the two coefficients of x^2 and y^2, which in this case is 20. Therefore, a = 20. To determine the length of the major diameter of an ellipse, we must first find the distance between its two foci, which is equal to 2a. This can be derived from the eccentricity (e) of the ellipse, which is given by the equation: e = √(1 - (b^2/a^2)) Where e = distance between foci/distance between center and focus. Substituting our known values, we get: e = √(1 - (b^2/20^2)) Since the length between the center and focus is equal to half of the major diameter (a = 2a/2 = a), we can rewrite the equation as: e = √(1 - (a^2/a^2)) Therefore, we get: e = √(1 - 1) = √0 = 0 This means that the distance between the two foci is also 0, and consequently, the major diameter of the given ellipse is simply 2a. Therefore, the length of the major diameter of the ellipse 20x^2+2x+y^2 = 1 is 2(20) = 40 units. Question 3 Jon is camping with his family. He wants to hike to the lake, go fishing, and hike back before 1:45 P.M. It will take 50 minutes to hike to the lake and 1 hour and 45 minutes to hike back. Jon wants to fish for 2 hours and 50 minutes. What is the latest time Jon can start the hike to the lake? Answer Based on the given information, we can calculate the latest time Jon can start the hike to the lake by considering the total time needed for each activity and the desired end time of 1:45 P.M. First, we need to determine the total time needed for the hike to the lake and back. Since it takes 50 minutes to hike to the lake and 1 hour and 45 minutes to hike back, the total time spent on hiking will be 50 minutes + 1 hour and 45 minutes = 2 hours and 35 minutes. Next, we need to calculate the total time that Jon wants to spend fishing, which is 2 hours and 50 minutes. Since this time is already longer than the total time needed for hiking, we can conclude that Jon must start hiking before he begins fishing. Therefore, to ensure that Jon is able to complete his planned activities and return before 1:45 P.M, he must start hiking to the lake before 1:45 P.M- 2 hours and 35 minutes = 11:10 AM. In other words, Jon must start hiking to the lake at or before 11:10 AM to have enough time to reach the lake, go fishing for 2 hours and 50 minutes, and hike back before 1:45 P.M. Question 4 Kendrick got in his car and started driving toward his grandmother's house. After driving for 55 minutes, Kendrick took a break for 1 hour and 10 minutes to have a snack. Then Kendrick drove 1 hour and 40 minutes and finally arrived at his grandmother's house at 11:35 A.M. What time did Kendrick start driving?. Answer Based on the given information, Kendrick started driving toward his grandmother's house at 9:50 A.M. Firstly, we need to analyze the information given to us. We know that Kendrick drove for 55 minutes before taking a break for 1 hour and 10 minutes. This means that he stopped driving at 11:05 A.M. Then he resumed driving for 1 hour and 40 minutes, which brings us to 12:45 P.M. Adding the break time of 1 hour and 10 minutes to this, Kendrick would have started driving at 11:35 A.M. However, in the question it is stated that he arrived at his grandmother's house at 11:35 A.M. This means that he must have started driving before 11:35 A.M. To determine the exact time Kendrick started driving, we need to subtract the total driving time (55 minutes + 1 hour and 40 minutes) from 11:35 A.M. This would give us the time Kendrick started driving, which is 9:50 A.M. Question 5 To find the x-intercepts, we set y to be zero and solve for x. For the given relationship y = -3(x – 1)(x + 5), the x-intercepts are x = 1 and x = -5. The equation for the vertex of a quadratic function in the form y = a(x - h)^2 + k is (h, k). In this case, the equation is y = -3(x - 1)(x + 5), so the vertex is (1, 0). To sketch the graph, we plot the x-intercepts at x = 1 and x = -5, and draw a parabola opening downwards with the vertex at (1, 0). can you show me how to draw the parabola thanks Answer In order to sketch the graph of a quadratic function, it is important to understand the properties and characteristics of a parabola. The general form of a quadratic function is y = ax2 + bx + c, where a, b, and c are constants. The coefficient "a" determines the direction and steepness of the parabola, while the constants "b" and "c" affect the position of the vertex. In this particular case, the given relationship y = -3(x – 1)(x + 5) is already in factored form, allowing us to easily identify the x-intercepts at x = 1 and x = -5. To find the vertex of the parabola, we can use the formula (h, k), where h is calculated as -b/2a and k is the value of y at that particular x-coordinate. Using the values of a = -3 and b = -4 from the given equation, we can calculate the vertex as follows: h = -(-4)/2(-3) = 4/(-6) = -2/3 k = -(3)(-2/3-1)(-2/3+5) = -(3)(-5/3)(13/3) = 65/9 Therefore, the vertex is at (-2/3, 65/9). To sketch the graph, we can now plot the x-intercepts at x = 1 and x = -5, and the vertex at (-2/3, 65/9). The parabola will open downwards as the coefficient "a" is negative. We can also plot a few more points to get a better visual of the shape of the parabola. For example, we can plug in x = 0 to get y = -15, and plot the point (0, -15).