Question

1. 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

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.