An electric device delivers a current of [tex]$15.0 A$[/tex] for 30 seconds. How many electrons flow through it?
Answer (2)
Substitute the given values of t and h into the equation h ( t ) = − 4.9 t 2 + h 0 to obtain three equations.
Solve each equation for h 0 .
Calculate h 0 from each equation: h 0 = 60 .
The quadratic equation that models the situation is h ( t ) = − 4.9 t 2 + 60 .
Explanation
Understanding the Problem We are given the function h ( t ) = − 4.9 t 2 + h 0 which describes the height of an object t seconds after it falls from an initial height h 0 . We are also given a table of data showing the height h of a pebble at different times t . Our goal is to find the value of h 0 that best fits the data and choose the correct quadratic equation.
Setting up the Equations We can use the data provided in the table to determine the initial height h 0 . We will substitute the given values of t and h into the equation h ( t ) = − 4.9 t 2 + h 0 to obtain three equations.
Solving for h0 using t=1 Using the first data point, t = 1 and h = 55.1 , we have: 55.1 = − 4.9 ( 1 ) 2 + h 0 55.1 = − 4.9 + h 0 Solving for h 0 , we get: h 0 = 55.1 + 4.9 = 60
Solving for h0 using t=2 Using the second data point, t = 2 and h = 40.4 , we have: 40.4 = − 4.9 ( 2 ) 2 + h 0 40.4 = − 4.9 ( 4 ) + h 0 40.4 = − 19.6 + h 0 Solving for h 0 , we get: h 0 = 40.4 + 19.6 = 60
Solving for h0 using t=3 Using the third data point, t = 3 and h = 15.9 , we have: 15.9 = − 4.9 ( 3 ) 2 + h 0 15.9 = − 4.9 ( 9 ) + h 0 15.9 = − 44.1 + h 0 Solving for h 0 , we get: h 0 = 15.9 + 44.1 = 60
Finding the Quadratic Equation Since all three data points give us the same value for h 0 , we can conclude that h 0 = 60 . Therefore, the quadratic equation that models the situation is: h ( t ) = − 4.9 t 2 + 60
Final Answer The quadratic equation that models the situation is h ( t ) = − 4.9 t 2 + 60 .
Examples
Understanding the trajectory of objects in free fall is crucial in many real-world applications, such as engineering, physics, and even sports. For example, engineers use these principles to design structures that can withstand the impact of falling objects, while athletes use them to optimize their performance in sports like basketball or diving. By modeling the height of a falling object as a function of time, we can predict its position at any given moment and make informed decisions based on this information. This knowledge is also vital in fields like aerospace engineering, where precise calculations are needed to ensure the safe landing of spacecraft.
A current of 15.0 A for 30 seconds results in a total charge of 450 coulombs. This charge corresponds to approximately 2.81 × 1 0 21 electrons flowing through the device. The calculation uses the formulas for current, charge, and the charge of an electron to arrive at this result.
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