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Igbo uploaded a funny video of his cat onto a website.
The relationship between the elapsed time,
d, in days, since the video was first uploaded, and the total number of views,
V(d), that the video received is modeled by the following function.

V(d)=10^(1.5 d)
How many days will it take for the video to receive
1,000,000 views? Round your answer, if necessary, to the nearest hundredth.
days

Igbo uploaded a funny video of his cat onto a website. $\newline$ The relationship between the elapsed time, $d$ , in days, since the video was first uploaded, and the total number of views, $V(d)$ , that the video received is modeled by the following function. $\newline$ $V(d)=10^{1.5 d}$ $\newline$ How many days will it take for the video to receive $1,000,000$ views? Round your answer, if necessary, to the nearest hundredth. $\newline$ days

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Write linear functions: word problems

Full solution

Q. Igbo uploaded a funny video of his cat onto a website.

\newline

The relationship between the elapsed time,

d

, in days, since the video was first uploaded, and the total number of views,

V(d)

, that the video received is modeled by the following function.

\newline

V(d)=10^{1.5 d}

\newline

How many days will it take for the video to receive

1,000,000

views? Round your answer, if necessary, to the nearest hundredth.

\newline

days

Identify function and target views: Identify the given function and the target number of views. $\newline$ The function given is $V(d) = 10^{(1.5d)}$ , which models the total number of views $V$ as a function of the number of days $d$ since the video was uploaded. We want to find the number of days it will take for the video to receive $1,000,000$ views.
Set up equation for $1$ , $000$ , $000$ views: Set up the equation to solve for the number of days $d$ when the video reaches $1$ , $000$ , $000$ views. $\newline$ We need to find the value of $d$ such that $V(d) = 1,000,000$ . So we set up the equation: $\newline$ $1,000,000 = 10^{(1.5d)}$
Solve equation using logarithms: Solve for $d$ by using logarithms. $\newline$ To solve for $d$ , we can take the logarithm of both sides of the equation. We will use the common logarithm (base $10$ ) since the right side of the equation is a power of $10$ . $\newline$ $\log(1,000,000) = \log(10^{1.5d})$
Apply properties of logarithms: Apply the properties of logarithms. $\newline$ Using the property that $\log(a^b) = b \cdot \log(a)$ , we can simplify the right side of the equation: $\newline$ $\log(1,000,000) = 1.5d \cdot \log(10)$ $\newline$ Since $\log(10)$ is $1$ , the equation simplifies to: $\newline$ $\log(1,000,000) = 1.5d$
Calculate logarithm of $1$ , $000$ , $000$ : Calculate the logarithm of $1$ , $000$ , $000$ . $\newline$ Using a calculator, we find that $\log(1,000,000) = 6$ because $10^6 = 1,000,000$ . $\newline$ So the equation becomes: $\newline$ $6 = 1.5d$
Solve for d: Solve for d. $\newline$ To find d, we divide both sides of the equation by $1.5$ : $\newline$ $d = \frac{6}{1.5}$ $\newline$ $d = 4$

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