An electric device delivers a current of [tex]$15.0 A$[/tex] for 30 seconds. How many electrons flow through it?
Answer (2)
Analyze the quadratic function y = − x 2 + 50 x − 60 to model nerve impulses.
Determine the x-coordinate of the vertex using x v = 2 a − b , which gives x v = 25 .
Substitute x = 25 into the equation to find the maximum firing rate.
Calculate the maximum firing rate as y v = 565 .
The maximum firing rate is 565 .
Explanation
Problem Analysis We are given the equation y = − x 2 + 50 x − 60 , which models the number of impulses fired by a nerve, where y is the number of responses per millisecond and x is the number of milliseconds since the nerve was stimulated. We want to find the maximum firing rate, which corresponds to the maximum value of the function.
Understanding the Equation The equation is a quadratic function in the form of a parabola. Since the coefficient of the x 2 term is negative ( − 1 ), the parabola opens downward, meaning it has a maximum point (vertex). To find the maximum firing rate, we need to find the vertex of the parabola.
Finding the Vertex x-coordinate The x-coordinate of the vertex of a parabola given by y = a x 2 + b x + c is given by the formula x v = 2 a − b . In our case, a = − 1 and b = 50 . So, we have:
x v = 2 ( − 1 ) − 50 = − 2 − 50 = 25
This means the maximum firing rate occurs at x = 25 milliseconds.
Finding the Vertex y-coordinate Now, we substitute x = 25 into the equation to find the maximum firing rate (the y-coordinate of the vertex):
y v = − ( 25 ) 2 + 50 ( 25 ) − 60 y v = − 625 + 1250 − 60 y v = 565
So, the maximum firing rate is 565 responses per millisecond.
Final Answer Therefore, the maximum firing rate is 565 responses per millisecond.
Examples
Understanding the maximum firing rate of neurons is crucial in neuroscience. For example, when studying reflexes, knowing the peak response time and intensity helps researchers understand the speed and strength of neural signals. This knowledge can be applied in designing treatments for nerve damage or neurological disorders, where optimizing neural firing rates can improve patient outcomes. Also, in sports science, understanding nerve response can help athletes improve reaction times.
In 30 seconds, approximately 2.81 × 1 0 21 electrons flow through a device delivering 15.0 A of current. This is calculated using the relationship between current, charge, and the charge of a single electron. The total charge from the current and time leads us to this large number of electrons.
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