13. the average monthly cell phone bill in a given year for a us consumer can be modeled by…

13. the average monthly cell phone bill in a given year for a us consumer can be modeled by (b(t)=0.23t^{2}-0.05t + 41), where (t) measures time in years since the start of 2020, and (b(t)) is the average billed amount, in dollars. for example, in 2020, the average cell phone bill was $41 a month. a. find (b(t)) (b(t)=) b. evaluate (b(10)) and use it to interpret in the context of the problem. (do not round.) the average monthly cell phone bill is by

13. the average monthly cell phone bill in a given year for a us consumer can be modeled by (b(t)=0.23t^{2}-0.05t + 41), where (t) measures time in years since the start of 2020, and (b(t)) is the average billed amount, in dollars. for example, in 2020, the average cell phone bill was $41 a month. a. find (b(t)) (b(t)=) b. evaluate (b(10)) and use it to interpret in the context of the problem. (do not round.) the average monthly cell phone bill is by

Answer

Explanation:

Step1: Apply power - rule for differentiation

The power - rule states that if $y = ax^n$, then $y^\prime=anx^{n - 1}$. For the function $B(t)=0.23t^{2}-0.05t + 41$, the derivative of $0.23t^{2}$ is $0.23\times2t^{2 - 1}=0.46t$, the derivative of $-0.05t$ is $-0.05\times1t^{1 - 1}=-0.05$, and the derivative of the constant 41 is 0. So, $B^\prime(t)=0.46t-0.05$.

Step2: Evaluate $B^\prime(10)$

Substitute $t = 10$ into $B^\prime(t)$. We get $B^\prime(10)=0.46\times10-0.05$. $B^\prime(10)=4.6 - 0.05=4.55$. In the context of the problem, $B^\prime(t)$ represents the rate of change of the average monthly cell - phone bill with respect to time (in years since 2020). So, $B^\prime(10) = 4.55$ means that 10 years after the start of 2020 (i.e., in 2030), the average monthly cell - phone bill is increasing by $$4.55$ per year.

Answer:

a. $B^\prime(t)=0.46t - 0.05$ b. The average monthly cell phone bill is increasing by $$4.55$ per year.