An electric device delivers a current of [tex]$15.0 A$[/tex] for 30 seconds. How many electrons flow through it?
Answer (2)
Set the two equations equal to each other: x + 2 = − x 2 + 2 .
Rearrange the equation to form a quadratic equation: x 2 + x = 0 .
Factor the quadratic equation: x ( x + 1 ) = 0 .
Solve for x and y : The solutions are ( 0 , 2 ) , ( − 1 , 1 ) .
Explanation
Understanding the Problem We are given a system of two equations with two variables, x and y . The equations are y = x + 2 and y = − x 2 + 2 . We need to find all ordered pairs ( x , y ) that satisfy both equations.
Setting the Equations Equal To solve this system of equations, we can set the two expressions for y equal to each other, since both expressions must be true for the same x and y values. This gives us the equation x + 2 = − x 2 + 2 .
Rearranging the Equation Now, we rearrange the equation to form a quadratic equation in x . Subtracting x + 2 from both sides gives 0 = − x 2 − x . Multiplying both sides by − 1 gives x 2 + x = 0 .
Factoring the Quadratic We can factor the quadratic equation as x ( x + 1 ) = 0 .
Solving for x To solve for x , we set each factor equal to zero: x = 0 or x + 1 = 0 . This gives us two possible values for x : x = 0 or x = − 1 .
Solving for y Now, we substitute each value of x back into either of the original equations to find the corresponding y values. Let's use the equation y = x + 2 . If x = 0 , then y = 0 + 2 = 2 . If x = − 1 , then y = − 1 + 2 = 1 .
The Solutions Therefore, the solutions are ( 0 , 2 ) and ( − 1 , 1 ) .
Examples
Systems of equations are used in many fields, such as physics, engineering, and economics. For example, in physics, you might have one equation that describes the position of an object as a function of time, and another equation that describes the velocity of the object as a function of time. By solving the system of equations, you can find the time and position at which the object has a certain velocity. In economics, you might have one equation that describes the supply of a product as a function of price, and another equation that describes the demand for the product as a function of price. By solving the system of equations, you can find the equilibrium price and quantity of the product.
Approximately 2.81 × 1 0 21 electrons flow through the device when a current of 15.0 A is delivered for 30 seconds. This is calculated by first determining the total charge that flows and then converting it to the number of electrons using the charge of a single electron. The relationship between current, time, and charge allows us to find this number effectively.
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