An online video platform tracks the cumulative number of views (in millions) that a…

MATHEMATICS · Multi-part · AP · Calculus BC · Differential Equations · Differential Equations

Problem

An online video platform tracks the cumulative number of views

V(t)

(in millions) that a newly released short film receives, where

t

is measured in hours since release. Marketing data suggest that the potential audience—the maximum number of users who could eventually watch the film—grows over time as the platform’s subscriber base increases. Let the time-dependent potential audience be

M(t)=2+0.05t

, measured in millions of users. The viewing data are modeled by the differential equation

\frac{dV}{dt}=rV\bigl(M(t)-V\bigr),\quad V(0)=0.10,

where

r

is a positive constant with units of

\text{hour}^{-1}

. For this film,

r=0.30\,\text{hr}^{-1}

. (a) Write the explicit expression for

\dfrac{dV}{dt}

in terms of

t

and

V

using the numerical values provided. (b) Explain qualitatively how the fact that

M(t)

increases with time affects the long-term behavior of

V(t)

. Your answer should address whether

V(t)

can approach a fixed horizontal asymptote. (c) Determine all values of

V

for which

\dfrac{dV}{dt}=0

. Classify each as stable or unstable, and justify your classification. (d) At

t=20

hours, estimate the instantaneous growth rate in views using the expression from part (a) and

V(20)=1.2

million. Interpret this value in context.

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