Reading and interpreting data #2 - Questions and Answers

Question 1 A jetliner can fly 6.00 hours on a full load of fuel. Without wind it flies at a speed of 2.40 X 10²m/s. The planes to make a round-trip by heading due west for a certain distance, turning around, and then heading due east for the return trip. During the entire flight, however, the place encounters a 57.8 m/s wind from the jet stream, which blows from west to east. What is the maximum distance that the plane can travel due west and just be able to return home? Answer 1 The jetliner can fly for 6 hrs on a load of fuel, so the whole trip would have to take 6 hours. So that is 6 hrs × (60 * 60) seconds / hr = 21600 seconds it can fly and at 120 m/s * 21600 s = 2592 km is how far the plane can travel in the round trip. let's see a diagram: departure flight wind = 57.8 m/s → plane 240 m/s → plane 240 m/s → wind 57.8 m/s return flight you can see that the wind counteracts the velocity of the plane on the departure flight so the velocity of the plane will be (240 - 57.8) = 182.2m/s on the way back, the wind and plane velocity will add up so the total velocity will be (240 + 57.8) = 297.8m/s so we can say: so on the way back the plane travels (297.8 - 182.2) = 115.6 m/s faster than on the departure. the distance both ways must be the same so: distance there = 182.2 m/s * t1 distance back = 297.8 m/s * t2 if we take the ratio of these 2: 297.8 / 182.2 = 1.6345 how many times faster the return time is that the departure. t₁ = 1.6345t₂. the total time (t₁ + t₂ must be 6 hrs or 21 600 seconds) so substitute in the above to get: (1.6345t₂) + t₂ = 21600 seconds. 2.6345t₂ = 21600 s t₂ = 8198.899 seconds. so t₁ must be 21600 - 8198.899 = 13401.1 seconds. now we can plug this into our distance formula above: distance there = 182.2 m/s * 13401.1 s = 2441680 m = 2.44 * 10⁶ m is the distance that the plane can travel Answer 2 Step 1: Let's denote the maximum distance traveled due west as d (in meters). When the plane flies due west, it will be flying against the wind, so its effective speed will be reduced. The plane's effective speed can be calculated as the difference between its actual speed and the wind speed: Veffective = V plane - Vwind where Vplane is the plane's speed without wind (given as 2.40×10² m/s) and Vwind is the wind speed (given as 57.8 m/s). Therefore, the effective speed when flying due west is: Veffective = (2.40×10²) - 57.8 m/s. The total time spent flying due west is the distance traveled divided by the effective speed: twest = d / Veffective Similarly, the total time spent flying due east is the distance traveled divided by the sum of the plane's speed and the wind speed: teast = d / ((2.40×10²) + 57.8) Since the total flying time is 6 hours, the sum of the times spent flying west and east must be equal to 6: twest + teast = 6. Substituting the expressions for twest and teast, we get: d / Veffective + d / ((2.40×10²) + 57.8) = 6. Simplifying the equation and solving for d, we can find the maximum distance traveled due west: d = 6 * Veffective / (1 + (Veffective / ((2.40×10²) + 57.8))) Step 2: Now we can substitute the given values for Veffective, solve the equation, and find the maximum distance traveled due west. d = 6 * (2.40×10² - 57.8) / (1 + ((2.40×10² - 57.8) / (2.40×10² + 57.8))) Simplifying the expression further: d = 6 * 182.2 / (1 + (182.2 / 297.8)) d = 1093.2 / (1 + 0.6113) d = 1093.2 / 1.6113 d ≈ 678.40 meters Therefore, the maximum distance that the plane can travel due west and still be able to return home is approximately 678.40 meters. Answer 3 When the plane is flying due west, the effective speed is the sum of the plane's speed and the speed of the wind blowing in the opposite direction. So, the effective speed while flying west is 2.40×10² m/s - 57.8 m/s. d The time taken to travel this distance can be calculated using the formula t = -, where d is the distance and v is the velocity. Thus, the time taken to travel x meters due west is given by: twest = x / (2.40×10² - 57.8) Since the total flight time is 6.00 hours, the time taken for the return trip is given by: teast = 6.00 - twest During the return trip, the effective speed while flying east is the sum of the plane's speed and the speed of the wind blowing in the same direction. So, the effective speed while flying east is 2.40×10² m/s + 57.8 m/s. Using the same formula as before, the distance traveled during the return trip is: deast = teast * (2.40×10² + 57.8) Since the plane needs to return home, the distance traveled due west and due east must be the same. Therefore, we can set x equal to deast: x = teast * (2.40×10² + 57.8) Now, we can substitute the value of teast from the earlier equation: x = (6.00 - twest) * (2.40×10² + 57.8) Substituting the value of twest from the first equation, we have: x = (6.00 - x/(2.40×10² - 57.8)) * (2.40×10² + 57.8) Now, we can solve this equation to find the value of x: x = (6.00 - x/(2.40×10² - 57.8)) * (2.40×10² + 57.8) Simplifying this equation will give us the maximum distance that the plane can travel due west and still be able to return home. Answer 4 Explanation: To find the maximum distance the plane can travel due west and still be able to return home, we need to consider the effect of the wind on the plane's speed. Let's denote the maximum distance the plane can travel due west as d (in meters). When the plane is flying due west, it will be flying against the wind, which will reduce its effective speed. The effective speed of the plane can be calculated by subtracting the wind speed from the plane's speed: Veffective = Vplane - Vwind Veffective = 2.40×10² m/s - 57.8 m/s During the westward trip, the plane will take d / Veffective hours to travel the distance d. Now, for the eastward trip, the plane will be flying with the wind, which will increase its effective speed. The effective speed for the eastward trip can be calculated by adding the wind speed to the plane's speed: Veffective = Vplane + Vwind Veffective = 2.40×10² m/s + 57.8 m/s During the eastward trip, the plane will take d / Veffective hours to travel the same distance d. Since the total flight time is given as 6.00 hours, the sum of the westward and eastward trip times must equal 6.00 hours: d / (2.40×10² - 57.8) + d / (2.40×10² + 57.8) = 6.00 Simplifying the equation: d * (1/ (2.40 * 10² - 57.8) + 1 / (2.40 * 10² + 57.8)) = 6.00 2d * (2.40 * 10²) / ((2.40 * 10²)² - (57.8)²) = 6.00 Solving for d: d = 6.00 * ((2.40 * 10²)² - (57.8)²) / (2 * (2.40 * 10²)) Substituting the values of Veffective: d = (6.00 * (2.40 * 10² - 57.8) * (2.40 * 10² + 57.8)) / (2 * (2.40 * 10²)) d ≈ 678.40 meters Therefore, the maximum distance the plane can travel due west and still be able to return home is approximately 678.40 meters. Question 2 How many zeros does a million and billion have. What is the equivalent lakhs and crore value of million and billion? Answer Result: One million is 1,000,000 (6 zeros) and one billion 1,000,000,000 (9 zeros) although sometimes in British english is considered 1,000,000,000,000 (12 zeros) like in other european countries. A lakh is equivalent to 100,000 decimal units and a crore 10,000,000 so: one million is 10 lakhs. one billion 100 crores or 10,000 lakhs Question 3 In a gymnastics competition, Paiges Answer Paige's score in word form will be: Thirty seven & Twenty-Five Thousandth Question 4 Interpreting z-scores: Complete the following statements using your knowledge about z- scores. a. If the data is weight, the z-score for someone who is overweight would be -positive -negative -zero b. If the data is IQ test scores, an individual with a negative z-score would have a -high IQ -low IQ -average IQ c. If the data is time spent watching TV, an individual with a z-score of zero would -watch very little TV -watch a lot of TV -watch the average amount of TV d. If the data is annual salary in the U.S and the population is all legally employed people in the U.S., the z-scores of people who make minimum wage would be -positive -negative -zero Answer Step 1 Standardized z-score: The standardized z-score represents the number of standard deviations the data point is away from the mean. If the z-score takes positive value when it is above the mean (0). If the z-score takes negative value when it is below the mean (0). (a) Interpreting the z-score for the statement " if the data is weight, the z-score for someone who is overweight". The z-score is interpreted below as follows: From the information, given the statement is "if the data is weight, the z-score for someone who is overweight" which indicates that the z-score represent the positive score since because the word overweight represents positive value. Correct option: Positive. Step 2 (b) Interpreting the z-score for the statement "If the data is IQ test scores, an individual with a negative z-score would have". The z-score is interpreted below as follows: From the information, given the statement If the data is IQ test scores, an individual with a negative z-score would have" which indicates that the z-score represent the negative score since because the word negative score represents low IQ value. Correct option: Low IQ. (c) Interpreting the z-score for the statement "If the data is time spent watching TV, then an individual with a z-score of zero would". The z-score is interpreted below as follows: From the information, given the statement "If the data is time spent watching TV, an individual with a z-score of zero would" which indicates that the z-score represent the average score since because the word 0 represents average amount of watching the TV.