the number of visitors p to a website in a given week over a 1 - year period is given by ( p(t)=121+(t…

the number of visitors p to a website in a given week over a 1 - year period is given by ( p(t)=121+(t - 88)e^{0.02t} ), where t is the week and ( 1leq tleq52 ).\na) over what interval of time during the 1 - year period is the number of visitors decreasing?\nb) over what interval of time during the 1 - year period is the number of visitors increasing?\nc) find the critical point, and interpret its meaning.\na) the number of visitors is decreasing over the interval ( (1,38) ).\n(simplify your answer. type integers or decimals rounded to three decimal places as needed. type your answer in interval notation.)\nb) the number of visitors is increasing over the interval ( (38,52) ).\n(simplify your answer. type integers or decimals rounded to three decimal places as needed. type your answer in interval notation.)\nc) the critical point is \n(type an ordered pair. type integers or decimals rounded to three decimal places as needed.)

the number of visitors p to a website in a given week over a 1 - year period is given by ( p(t)=121+(t - 88)e^{0.02t} ), where t is the week and ( 1leq tleq52 ).\na) over what interval of time during the 1 - year period is the number of visitors decreasing?\nb) over what interval of time during the 1 - year period is the number of visitors increasing?\nc) find the critical point, and interpret its meaning.\na) the number of visitors is decreasing over the interval ( (1,38) ).\n(simplify your answer. type integers or decimals rounded to three decimal places as needed. type your answer in interval notation.)\nb) the number of visitors is increasing over the interval ( (38,52) ).\n(simplify your answer. type integers or decimals rounded to three decimal places as needed. type your answer in interval notation.)\nc) the critical point is \n(type an ordered pair. type integers or decimals rounded to three decimal places as needed.)

Answer

Explanation:

Step1: Find the derivative of (P(t))

Given (P(t)=121+(t - 88)e^{0.02t}). Using the product rule ((uv)^\prime=u^\prime v+uv^\prime), where (u=t - 88), (u^\prime = 1) and (v=e^{0.02t}), (v^\prime=0.02e^{0.02t}). (P^\prime(t)=e^{0.02t}+0.02(t - 88)e^{0.02t}=e^{0.02t}(1 + 0.02(t - 88))=e^{0.02t}(0.02t-1.76 + 1)=e^{0.02t}(0.02t - 0.76))

Step2: Find the critical point

Set (P^\prime(t)=0). Since (e^{0.02t}>0) for all (t), we solve (0.02t-0.76 = 0). (0.02t=0.76), then (t=\frac{0.76}{0.02}=38) Substitute (t = 38) into (P(t)): (P(38)=121+(38 - 88)e^{0.02\times38}=121-50e^{0.76}\approx121-50\times2.138\approx121 - 106.9=14.1)

Answer:

The critical point is ((38,14.1)). It means that at week (t = 38), the rate of change of the number of website visitors changes from negative (visitors were decreasing) to positive (visitors start increasing). So, it is the point where the number of visitors reaches a local minimum.