Question

NO LINKS!! NOT MULTIPLE CHOICE!!!

56. Kotkit is the hottest new social media app. It is lowkey fire. It slays! There were no crumbs left. Users post their Tiktoks to Kotkit, but play them backwards. At the start of the year, there are 1500 users, but the number of users grows exponentially at a rate of 12% per day.

a. Write a function K(t) that expresses the number of users t days after the start of the year.

b. How many users are of the Kotkit after 30 days?

.c After how many days will the numbers of users reach one million users?

Answer 1

`\textsf{a)} \quad K(t)=1500(1.12)^t`

b) 44940

c) 57.4 days

Explanation:

Exponential Growth Function

`f(t)=a(1+r)^t`

where:

• a is the initial value (y-intercept)
• r is the growth rate (in decimal form)
• t is the time

Given values:

• a = 1500
• r = 12% per day = 0.12
• t = time in days

Part a

To create a function K(t) that expresses the number of users t days after the start of the year, substitute the given values into the exponential growth formula:

`\implies K(t)=1500(1+0.12)^t`

`\implies K(t)=1500(1.12)^t`

Part b

To calculate how many users there are after 30 days, substitute t = 30 into the function K(t) from part a:

`\implies K(30)=1500(1.12)^(30)`

`\implies K(30)=44939.8831...`

`\implies K(30)=44940\; \rm users`

Part c

To calculate after how many days it takes to reach one million users, substitute K(t) = 1,000,000 into the function and solve for t:

`\implies 1000000=1500(1.12)^t`

`\implies (2000)/(3)=1.12^t`

`\implies \ln \left((2000)/(3)\right)=\ln 1.12^t`

`\implies \ln \left((2000)/(3)\right)=t \ln 1.12`

`\implies t=(\ln \left((2000)/(3)\right))/(\ln 1.12)`

`\implies t=57.3755016...`

`\implies t=57.4\; \rm days`