The amount of pollution, A (in part per million), present in the air outside a factory on a recent weekday is represented by the equation A= -t^2+16t+6, where t is the number of hours after 7 AM.

(a) find the Times (in radical form) when there were 42 parts per million of pollution.
(b) use your answer to approximate the Times to the nearest hour.
(c) do you think the pollution level at noon would be higher or lower than 42 parts per million? Explain

(a) just solve for t in

-t^2 + 16t + 6 = 42

(b) just add that many hours to 7:00 AM

(c) pollution at noon is A(5) = 61
So, looks like it is.

To find the times when there were 42 parts per million of pollution, we will substitute A = 42 into the given equation A = -t^2 + 16t + 6 and solve for t.

(a) Substituting A = 42 into the equation, we have:
42 = -t^2 + 16t + 6.

Rearranging the equation to a quadratic form:
t^2 - 16t - 36 = 0.

Now, we need to solve this quadratic equation. We can either factor it or use the quadratic formula. Since factoring may not be straightforward in this case, let's use the quadratic formula:

The quadratic formula is given by:
t = (-b ± √(b^2 - 4ac)) / (2a).

For our equation t^2 - 16t - 36 = 0, the coefficients are:
a = 1, b = -16, and c = -36.

Substituting these values into the quadratic formula, we have:
t = (-(-16) ± √((-16)^2 - 4(1)(-36))) / (2(1)).

Simplifying further:
t = (16 ± √(256 + 144)) / 2.
t = (16 ± √400) / 2.
t = (16 ± 20) / 2.

Simplifying the expression:
t = (16 + 20) / 2 or t = (16 - 20) / 2.
t = 36 / 2 or t = -4 / 2.
t = 18 or t = -2.

Hence, the times (in radical form) when the pollution level was 42 parts per million are t = 18 and t = -2.

(b) To approximate the times to the nearest hour, we round the values of t to the nearest whole number. Therefore, t = 18 would be approximately 18 hours after 7 AM, and t = -2 would be approximately 2 hours before 7 AM. Hence, the approximate times are 1 PM and 5 AM.

(c) The pollution level at noon is t = 12 hours after 7 AM. Substituting t = 12 into the given equation A = -t^2 + 16t + 6, we can find the pollution level at noon. By finding A(12), we can determine whether it is higher or lower than 42 parts per million.