A researcher in physiology has decided that a good mathematical model for the number of impulses fired after a nerve has been stimulated is given by $y=-x^2+50 x-60$, where $y$ is the number of responses per millisecond and $x$ is the number of milliseconds since the nerve was stimulated.
When will the maximum firing rate be reached?
Answer (1)
Analyze the quadratic function y = − x 2 + 50 x − 60 to recognize it represents a downward-opening parabola.
Apply the vertex formula x = − 2 a b to find the x-coordinate of the vertex, which represents the time of maximum firing rate.
Substitute a = − 1 and b = 50 into the formula: x = − 2 ( − 1 ) 50 .
Calculate the time at which the maximum firing rate is reached: 25 .
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 time x when the maximum firing rate is reached.
Identifying the Quadratic Function The equation is a quadratic function in the form of y = a x 2 + b x + c , where a = − 1 , b = 50 , and c = − 60 . Since a < 0 , the parabola opens downward, meaning it has a maximum point (vertex).
Vertex Formula The x-coordinate of the vertex of a parabola given by y = a x 2 + b x + c is found using the formula x = − 2 a b . In our case, a = − 1 and b = 50 .
Calculating the Time Substituting the values of a and b into the vertex formula, we get:
x = − 2 ( − 1 ) 50 = − − 2 50 = 25
Thus, the maximum firing rate is reached at x = 25 milliseconds.
Final Answer Therefore, the maximum firing rate will be reached at 25 milliseconds.
Examples
Understanding the time at which a nerve reaches its maximum firing rate is crucial in various fields. For instance, in sports science, it can help optimize training regimens by timing stimuli to coincide with peak nerve response. Similarly, in medical treatments, knowing when a nerve's response is at its maximum can aid in delivering targeted therapies or interventions, enhancing their effectiveness and minimizing potential side effects. This concept is also applicable in designing prosthetic limbs that respond optimally to nerve signals, improving the user's control and dexterity.
