Question

A study conducted by a local cable company estimates that the number of subscribers will grow at the rate of 120+280t^7/4 new subscribers per month, t months from the start date of service. If there were 800 subscribers signed up after one month:

a) Find the function for the number of subscribers during any month t.

b) How many subscribers were signed up after 16 months?

Answer 1

Solution:

The rate of increase of the number of cable TV subscriber is $\small 120+280t^{\frac{7}{4}}$ where $\small t$ is the month from the start date of service.

The subscriber of Cable TV after one month is $\small 800$ .

a)

Integrate the given expression for cable TV with respect to $\small t$ we get

f (t) = (120 + 280 ) dt

$\small =120t+\frac{1120}{11}t^{\frac{11}{4}}+C$

For  $\small t=1, \: \: f\left ( 1 \right )=800$

$\small \therefore f\left ( 1 \right )=800=120+\frac{1120}{11}+C$

$\small \therefore C=800-120-\frac{1120}{11}=\frac{6360}{11}$

Thus, the function for the number of subscriber during any month $\small t$ is

$\small f\left ( t \right )=120t+\frac{1120}{11}t^{\frac{11}{4}}+\frac{6360}{11}$

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b)

For the second part put  $\small t=16$

$\small f\left ( 16 \right )=120\times 16+\frac{1120}{11}\left ( 16 \right )^{^{\frac{11}{4}}}+\frac{6360}{11}\approx 211022$

Thus , the number of subscriber signed up after $\small 16$ months is $\small 211022$ .