Research reports indicate that surveillance cameras at major intersections dramatically reduce the number of drivers who barrel

Question

2 months ago

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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

Mathematics

1 answer:

babunello [11.8K] · Answer 1 · 2026-03-20T06:47:13+03:00

Answer:

(a)Increasing

(b)t=1.34 years

(c)16 cameras annually

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 considered to be increasing when x₁≤x₂ leads to f(x₁)≤f(x₂).

Since in the interval (0,4), N(0)<n we="" conclude="" that="" the="" function="" is="" on="" rise.="">

(b) The point at which the number of communities employing surveillance cameras at intersections changed least rapidly corresponds to where the derivative of the function equals zero.

N(t) = 5.85t³-23.43t²+45.06t+69.5

N'(t)=17.49t²-46.86t+45.06

Setting N'(t)=0,

17.49t²-46.86t+45.06=0

Solving the quadratic equation yields values of t as:

t=1.3396-0.8842i

t=1.3396+0.8842i

Taking the real part gives us our minimum value,[

The moment at which the number of communities utilizing surveillance cameras at intersections changed least rapidly is:

t=1.34 (rounded to two decimal places)

(c)Rate of Increase through security cameras per 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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