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The number y of hits a new website receives each month can be modeled by y = 4060e^kt, where t...

Question:

The number y of hits a new website receives each month can be modeled by y = 4060e{eq}^{kt} {/eq}, where t represents the number of months the website has been operating. In the website's third month, there were 9,000 hits.

Find the value of k. (Round answer to four decimal places.)

Hits!

A hit is a terminology that is frequently seen on a website. It basically refers to the number of files downloaded from a website. These include photos, videos, documents, and applications.

Answer and Explanation:

Given:

{eq}t=3\\[0.2cm] y(3)=9000\\[0.2cm] {/eq}

We rearrange the equation to find the value of k.

$$y=4060e^{kt}\\[0.2cm] \dfrac{y}{4060}=e^{kt}\\[0.2cm] \ln \left| \dfrac{y}{4060} \right| = \ln \mid e^{kt} \mid \\[0.2cm] \ln \left| \dfrac{y}{4060} \right| =kt\\[0.2cm] k=\dfrac{\ln \left| \frac{y}{4060} \right|}{t}\\[0.2cm] $$

We now substitute the values of {eq}y {/eq} and {eq}t {/eq}.

$$k=\dfrac{\ln \left| \frac{9000}{4060} \right|}{3}\\[0.2cm] k\approx \dfrac{0.796}{3}\\[0.2cm] \color{red}{k\approx 0.265}\\[0.2cm] $$

Therefore, the equation becomes

$$y(t)=4060e^{0.265t}\\[0.2cm] $$


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