The Political Noise Problem. The amount of background
noise is important to television news reporters.
One station developed the formula N=-t^2+12t+54
showing the noise level in decibels (N) as it relates to the
time after the speaker stops talking in seconds (t). How
many seconds after the speaker stops will the noise level
be the greatest? Write and tell how you decided
To find the time after the speaker stops when the noise level is at its peak, we need to determine the value of 't' that corresponds to the highest point on the graph of the given quadratic equation.
The given formula is N = -t^2 + 12t + 54, where N represents the noise level in decibels and t represents the time after the speaker stops talking in seconds.
The formula represents a quadratic equation in standard form, where the noise level is the dependent variable (N) and time is the independent variable (t).
To find the time when the noise level is at its maximum, we can use the concept of the vertex of a parabola. The vertex represents the highest or lowest point on the graph of a quadratic equation.
The x-coordinate of the vertex of a quadratic equation in the standard form ax^2 + bx + c is given by the formula: x = -b / (2a).
In our case, the equation is N = -t^2 + 12t + 54, which matches the standard form. By comparing coefficients, we can see that a = -1, b = 12, and c = 54.
Using the formula, we can find the time when the noise level is at its peak:
t = -b / (2a)
t = -12 / (2*(-1))
t = -12 / (-2)
t = 6
Therefore, the noise level will be at its maximum 6 seconds after the speaker stops talking.
Explanation steps:
1. Understand the problem: The problem states that a television station has developed a formula to calculate the noise level in decibels as it relates to the time after the speaker stops talking.
2. Interpret the given information: The formula N = -t^2 + 12t + 54 represents the noise level (N) as a function of time (t).
3. Determine the objective: We need to find the time when the noise level is at its maximum.
4. Recall the formula for finding the vertex of a quadratic equation: The x-coordinate of the vertex is given by x = -b / (2a).
5. Identify the coefficients of the quadratic equation: In our case, a = -1, b = 12, and c = 54.
6. Apply the formula to find the time when the noise level is at its maximum: t = -b / (2a).
7. Simplify the equation to find the value of t.
8. Answer the question: The noise level will be at its maximum 6 seconds after the speaker stops talking.