the number of visitors ( p ) to a website in a given week over a 1 - year period is given by ( p(t)=117+(t…

the number of visitors ( p ) to a website in a given week over a 1 - year period is given by ( p(t)=117+(t - 87)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 (square) (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 (square) (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 (square) (type an ordered pair. type integers or decimals rounded to three decimal places as needed.)\ninterpret what the critical point means.\nthe critical point means that the number of visitors was (square) in week (square) at approximately (square) visitors. (round to the nearest integer as needed.)

the number of visitors ( p ) to a website in a given week over a 1 - year period is given by ( p(t)=117+(t - 87)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 (square) (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 (square) (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 (square) (type an ordered pair. type integers or decimals rounded to three decimal places as needed.)\ninterpret what the critical point means.\nthe critical point means that the number of visitors was (square) in week (square) at approximately (square) visitors. (round to the nearest integer as needed.)

Answer

Explanation:

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

Use the product rule ((uv)^\prime = u^\prime v+uv^\prime), where (u = t - 87), (u^\prime=1), (v = e^{0.02t}), (v^\prime=0.02e^{0.02t}). [ \begin{align*} P^\prime(t)&=(t - 87)^\prime e^{0.02t}+(t - 87)(e^{0.02t})^\prime\ &=e^{0.02t}+0.02(t - 87)e^{0.02t}\ &=e^{0.02t}(1 + 0.02t-1.74)\ &=e^{0.02t}(0.02t - 0.74) \end{align*} ] Since (e^{0.02t}>0) for all (t), we only consider (0.02t - 0.74).

Step2: Find where (P^\prime(t)<0) (decreasing)

Set (0.02t - 0.74<0), then (0.02t<0.74), (t < 37). Given (1\leq t\leq52), the number of visitors is decreasing when (t\in[1,37)).

Step3: Find where (P^\prime(t)>0) (increasing)

Set (0.02t - 0.74>0), then (0.02t>0.74), (t>37). Given (1\leq t\leq52), the number of visitors is increasing when (t\in(37,52]).

Step4: Find the critical point

Set (P^\prime(t) = 0), since (e^{0.02t}\neq0), then (0.02t-0.74 = 0), (t = 37). Substitute (t = 37) into (P(t)): [ \begin{align*} P(37)&=117+(37 - 87)e^{0.02\times37}\ &=117-50e^{0.74}\ &\approx117-50\times2.096\ &=117 - 104.8\ &=12.2 \end{align*} ] The critical point is ((37,12))

Answer:

a) ([1,37)) b) ((37,52]) c) ((37,12)); The critical point means that the number of visitors was minimum in week (37) at approximately (12) visitors.