An electric device delivers a current of [tex]$15.0 A$[/tex] for 30 seconds. How many electrons flow through it?
Answer (2)
Determine that the function has a minimum value because the coefficient of the x 2 term is positive.
Calculate the x-coordinate of the vertex using the formula x = − b / ( 2 a ) , which gives x = 5 .
Substitute x = 5 into the function to find the minimum value, which is f ( 5 ) = − 76 .
State the domain as ( − ∞ , ∞ ) and the range as [ − 76 , ∞ ) . The minimum value is − 76 and it occurs at x = 5 .
Explanation
Problem Analysis We are given the quadratic function f ( x ) = 3 x 2 − 30 x − 1 . We need to determine if it has a minimum or maximum value, find that value and where it occurs, and identify the domain and range of the function.
Minimum or Maximum To determine if the function has a minimum or maximum, we look at the coefficient of the x 2 term. Since the coefficient is 3, which is positive, the parabola opens upwards, and the function has a minimum value.
Finding the x-coordinate of the vertex To find the x-coordinate of the vertex (where the minimum occurs), we use the formula x = − b / ( 2 a ) , where a = 3 and b = − 30 . Thus, x = − ( − 30 ) / ( 2 ∗ 3 ) = 30/6 = 5 .
Finding the minimum value Now, we substitute x = 5 into the function to find the minimum value: f ( 5 ) = 3 ( 5 2 ) − 30 ( 5 ) − 1 = 3 ( 25 ) − 150 − 1 = 75 − 150 − 1 = − 76 .
Domain and Range The domain of a quadratic function is all real numbers, so the domain is ( − ∞ , ∞ ) . Since the function has a minimum value of -76, the range is [ − 76 , ∞ ) .
Final Answer Therefore, the function has a minimum value of -76, which occurs at x = 5 . The domain is ( − ∞ , ∞ ) , and the range is [ − 76 , ∞ ) .
Examples
Imagine you are designing a parabolic mirror for a solar oven. Knowing the vertex of the parabola (the minimum point in this case) is crucial for focusing sunlight efficiently. By finding the minimum point of a quadratic function, you can optimize the design of the mirror to maximize heat concentration and cooking performance. This principle extends to various applications, such as satellite dishes and acoustic reflectors, where focusing energy at a specific point is essential.
Approximately 2.81 × 1 0 21 electrons flow through the electric device when a current of 15.0 A is delivered for 30 seconds. This is calculated by determining the total charge and then dividing by the charge of a single electron. Knowing the relationship between current, charge, and time is essential for this calculation.
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