SOLUTION: 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

Algebra ->  Quadratic Equations and Parabolas -> SOLUTION: 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      Log On


   

Question 207774: 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?
i would be sooo greatful if someone could give me some steps and tools to do this problem!!! thankyou in advance!!!
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
It turns out that the max decibel level N will occur at the vertex. So we need to find the vertex.

Note: the vertex is the highest/lowest point on a parabola. Also, I'm going to use 'x' in place of 't' and 'y' in place of 'N'




In order to find the vertex, we first need to find the x-coordinate of the vertex.


To find the x-coordinate of the vertex, use this formula: x=%28-b%29%2F%282a%29.


x=%28-b%29%2F%282a%29 Start with the given formula.


From y=-x%5E2%2B12x%2B54, we can see that a=-1, b=12, and c=54.


x=%28-%2812%29%29%2F%282%28-1%29%29 Plug in a=-1 and b=12.


x=%28-12%29%2F%28-2%29 Multiply 2 and -1 to get -2.


x=6 Divide.


So the x-coordinate of the vertex is x=6. Note: this means that the axis of symmetry is also x=6.


Now that we know the x-coordinate of the vertex, we can use it to find the y-coordinate of the vertex.


y=-x%5E2%2B12x%2B54 Start with the given equation.


y=-%286%29%5E2%2B12%286%29%2B54 Plug in x=6.


y=-1%2836%29%2B12%286%29%2B54 Square 6 to get 36.


y=-36%2B12%286%29%2B54 Multiply -1 and 36 to get -36.


y=-36%2B72%2B54 Multiply 12 and 6 to get 72.


y=90 Combine like terms.


So the y-coordinate of the vertex is y=90.


So the vertex is .


The min/max is simply the y-coordinate of the vertex. In this case, the max is 90.


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Answer:


So the highest noise level is 90 decibels (which occurs when t=6 ie when the time is 6 seconds).