Research reports indicate that surveillance cameras at major intersections dramatically reduce the number of drivers who barrel through red lights. The cameras automatically photograph vehicles that drive into intersections after the light turns red. Vehicle owners are then mailed citations instructing them to pay a fine or sign an affidavit that they weren't driving at the time. The function N(t) = 5.85t^3 - 23.43t^2 + 45.06t + 69.5 (0 lessthanorequalto t lessthanorequalto 4) gives the number, N(t), of communities in an undetermined first-world country using surveillance cameras at intersections in year t, with t = 0 corresponding to 2003. (a) Is N increasing or decreasing on (0, 4)? increasing decreasing varies(b) When was the number of communities using surveillance cameras at intersections changing least rapidly? (Round your answers to two decimal places.) t = What is the rate of increase? communities using a security camera/year

(a)Increasing

(b)t=1.34 years

(c)16 cameras per year

Step-by-step explanation:

Given the function

N(t) = 5.85t³-23.43t²+45.06t+69.5, 0≤t≤4

(a)N(0)=5.85(0)³-23.43(0)²+45.06(0)+69.5=69.5

N(4)=5.85(4)³-23.43(4)²+45.06(4)+69.5=249.26

A function is increasing whenever x₁≤x₂, f(x₁)≤f(x₂).

Since in the interval (0,4), N(0)<N(4), we say the function is increasing.

(b)The number of communities using surveillance cameras at intersections changed least rapidly at the point where the derivative of the function is zero.

N(t) = 5.85t³-23.43t²+45.06t+69.5

N'(t)=17.49t²-46.86t+45.06

If N'(t)=0,

17.49t²-46.86t+45.06=0

Solving the quadratic equation gives the values of t as:

t=1.3396-0.8842i

t=1.3396+0.8842i

We take the Real Part as our Minimum value,

The time when number of communities using surveillance cameras at intersections changed least rapidly is:

t=1.34(to 2 decimal places)

(c)Rate of Increase using a security camera/year.

N'(t)=17.49t²-46.86t+45.06

N'(t)=17.49(1)²-46.86(1)+45.06

=15.69

≈16 cameras/year

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