An electric device delivers a current of [tex]$15.0 A$[/tex] for 30 seconds. How many electrons flow through it?
Answer (2)
A device delivering 15.0 A for 30 seconds allows approximately 2.81 × 1 0 21 electrons to flow through it. We calculated this by first determining the total charge using the formula Q = I × t and then converting that charge into the number of electrons. Each electron has a charge of about 1.6 × 1 0 − 19 C .
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Calculate the time when the object hits the ground according to the line of best fit: x = 21.962 114.655 ≈ 5.22 seconds.
Compare this time with the actual time (4.6 seconds) to find the difference: 5.22 − 4.6 = 0.62 seconds.
Analyze the other statements and determine that they are false based on the given data and the line of best fit.
Conclude that the first statement is the most accurate: the object would have hit the ground approximately 0.6 seconds later according to the line of best fit. According to the line of best fit, the object would have hit the ground 0.6 seconds later than the actual time the object hit the ground.
Explanation
Problem Analysis We are given a table of data showing the height of a falling object at different times, and a line of best fit for the data: h = − 21.962 x + 114.655 , where h is the height in meters and x is the time in seconds. We need to determine which of the given statements best compares the line of best fit with the actual data.
Analyzing the Statements Let's analyze each statement:
Statement 1: According to the line of best fit, the object would have hit the ground 0.6 seconds later than the actual time the object hit the ground. To check this, we need to find the time when the line of best fit predicts the object hits the ground (i.e., when h = 0 ). 0 = − 21.962 x + 114.655 x = 21.962 114.655 ≈ 5.22 seconds The actual time the object hit the ground is 4.6 seconds. The difference is 5.22 − 4.6 = 0.62 seconds. So, the line of best fit predicts the object hits the ground approximately 0.62 seconds later than the actual time. This statement seems plausible.
Statement 2: According to the line of best fit, the object was dropped from a lower height. The initial height according to the line of best fit is when x = 0 : h = − 21.962 ( 0 ) + 114.655 = 114.655 meters. The actual initial height is 100 meters. Since 100"> 114.655 > 100 , the line of best fit predicts a higher initial height, not a lower one. So, this statement is false.
Statement 3: The line of best fit correctly predicts that the object reaches a height of 40 meters after 3.5 seconds. Let's find the height at x = 3.5 seconds using the line of best fit: h = − 21.962 ( 3.5 ) + 114.655 = − 76.867 + 114.655 = 37.788 meters. The actual height at 3.5 seconds is 40 meters. The line of best fit predicts 37.788 meters, which is not correct. So, this statement is false.
Statement 4: The line of best fit predicts a height of 4 meters greater than the actual height for any time given in the table. Let's check a few points from the table:
At x = 0 , the line of best fit gives h = 114.655 meters, while the actual height is 100 meters. The difference is 114.655 − 100 = 14.655 meters.
At x = 0.5 , the line of best fit gives h = − 21.962 ( 0.5 ) + 114.655 = 103.674 meters, while the actual height is 98.8 meters. The difference is 103.674 − 98.8 = 4.874 meters.
At x = 1.0 , the line of best fit gives h = − 21.962 ( 1.0 ) + 114.655 = 92.693 meters, while the actual height is 95.1 meters. The difference is 92.693 − 95.1 = − 2.407 meters. The difference is not consistently 4 meters. So, this statement is false.
Conclusion Based on our analysis, the first statement is the most accurate. The line of best fit predicts that the object would have hit the ground approximately 0.6 seconds later than the actual time.
Examples
Understanding lines of best fit is crucial in many real-world applications. For instance, in weather forecasting, meteorologists use historical data to create models that predict future temperatures. These models are essentially lines of best fit through past temperature data. Similarly, in economics, analysts use regression models to predict stock prices based on various economic indicators. These models help in making informed decisions about investments and resource allocation. The ability to interpret and compare these models with actual data is essential for making accurate predictions and informed decisions.
