Arthur sent a chain letter to his friends, asking them to forward the letter to more friends. The relationship between the elapsed time ttt, in months, since Arthur sent the letter, and the number of people, P(t)P(t)P, left parenthesis, t, right parenthesis, who receive the email is modeled by the following function: P(t)=12⋅5t3.6 Complete the following sentence about the growth rate of people who receive the email. The number of people who receive the email grows by a factor of 555 every months.

3.6

Step-by-step explanation:

Thinking about the problem

The modeling function is of the form P(t)=A⋅Bf(t), where B=5B=5B, equals, 5 and f(t)=\dfrac{t}{3.6}f(t)=  

3.6

t

f, left parenthesis, t, right parenthesis, equals, start fraction, t, divided by, 3, point, 6, end fraction.

Note that each time f(t)f(t)f, left parenthesis, t, right parenthesis increases by 111, the quantity is multiplied by B=5B=5B, equals, 5.

Therefore, we need to find the ttt-interval over which f(t)f(t)f, left parenthesis, t, right parenthesis increases by 111.

Hint #22 / 3

Finding the appropriate unit interval

fff is a linear function whose slope is \dfrac{1}{3.6}  

3.6

1

start fraction, 1, divided by, 3, point, 6, end fraction.

This means that whenever ttt increases by \Delta tΔtdelta, t, f(t)f(t)f, left parenthesis, t, right parenthesis increases by \dfrac{\Delta t}{3.6}  

3.6

Δt

start fraction, delta, t, divided by, 3, point, 6, end fraction.

Therefore, for f(t)f(t)f, left parenthesis, t, right parenthesis to increase by 111, we need \Delta t=3.6Δt=3.6delta, t, equals, 3, point, 6. In other words, the ttt-interval we are looking for is 3.63.63, point, 6 months.

Hint #33 / 3

Summary

The number of people who receive the email grows by a factor of 555 every 3.63.63, point, 6 months.

The slope intercept is easiest for me to use so...

y=mx+b

b= doesn't matter

m=2/3

y=2/3x+b

multiply the entire equation by 3 to cancel out the /3

3y=2x+3b

The value of b still doesn't matter, so ignore the 3b.

The equation that matches this is 3y=2x+6

Hope that helps