The Political Noise Problem. The amount of background noise is important to television news reporters. One station developed the formula
N = -t2 + 10t + 56
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 determine how many seconds after the speaker stops the noise level will be the greatest, we need to find the maximum value of the equation N = -t^2 + 10t + 56.
The formula represents a quadratic function in the form of N = at^2 + bt + c, where a = -1, b = 10, and c = 56.
In a quadratic function, the maximum or minimum value occurs at the vertex of the parabola. To find the vertex, we can use the formula t = -b/2a.
Substituting the values of a and b into the formula, we get:
t = -10 / (2 * -1)
t = -10 / -2
t = 5
Therefore, the noise level will be the greatest 5 seconds after the speaker stops.
To verify this, we can substitute t = 5 into the equation N = -t^2 + 10t + 56:
N = -(5)^2 + 10(5) + 56
N = -25 + 50 + 56
N = 81
So, 5 seconds after the speaker stops, the noise level will be the greatest at 81 decibels.