Gottfried wanted to see how contagious yawning can be. To better understand this, he conducted a social experiment by yawning in front of a random large crowd and observing how many people yawned as a result.
The relationship between the elapsed time ttt, in minutes, since Gottfried yawned, and the number of people in the crowd, P_{\text{minute}}(t)P
minute

(t)P, start subscript, start text, m, i, n, u, t, e, end text, end subscript, left parenthesis, t, right parenthesis, who yawned as a result is modeled by the following function:
Pminute(t)=5⋅(1.03)t
Complete the following sentence about the hourly rate of change in the number of people who yawn in Gottfried's experiment.
Round your answer to two decimal places.
Every hour, the number of people who yawn in Gottfried's experiment grows by a factor of

Every hour, the number of people who yawn in Gottfried's experiment grows by a factor of 5.89

Step-by-step explanation:

Understanding the problem

The expression for P_{\text{minute}}(t)P  

minute

(t)P, start subscript, start text, m, i, n, u, t, e, end text, end subscript, left parenthesis, t, right parenthesis, the number of people who yawn in Gottfried's experiment after ttt minutes, is 5⋅(1.03)t. This means that every minute, the number of people who yawn in Gottfried's experiment grows by a factor of 1.031.031, point, 03.

Let's change this expression so that the base tells us by which factor the number of people who yawn in Gottfried's experiment grows every hour.

Hint #22 / 3

Changing the unit

We want to find the expression for P_{\text{hour}}(x)P  

hour

(x)P, start subscript, start text, h, o, u, r, end text, end subscript, left parenthesis, x, right parenthesis, which models the number of people who yawn in Gottfried's experiment after xxx hours.

NOTE: The function P_{\text{hour}}P  

hour

P, start subscript, start text, h, o, u, r, end text, end subscript is not equivalent to P_{\text{minute}}P  

minute

P, start subscript, start text, m, i, n, u, t, e, end text, end subscript, although they model the same situation. [Tell me more.]

P_{\text{hour}}  

P, start subscript, start text, h, o, u, r, end text, end subscriptP_{\text{minute}}  

P, start subscript, start text, m, i, n, u, t, e, end text, end subscript

P_{\text{hour}}(1)  

P, start subscript, start text, h, o, u, r, end text, end subscript, left parenthesis, 1, right parenthesis11P_{\text{minute}}(1)  

P, start subscript, start text, m, i, n, u, t, e, end text, end subscript, left parenthesis, 1, right parenthesis11

Since there are \blueD{60}60start color #11accd, 60, end color #11accd minutes in an hour, t=\blueD{60}xt=60xt, equals, start color #11accd, 60, end color #11accd, x, and the expression for P_{\text{hour}}(x)P  

hour

(x)P, start subscript, start text, h, o, u, r, end text, end subscript, left parenthesis, x, right parenthesis is the same as the following expression.

Pminute(60x)=5⋅(1.03)60x=5⋅((1.03)60)x

Evaluating \maroonC{(1.03)^{60}}(1.03)  

60

start color #ed5fa6, left parenthesis, 1, point, 03, right parenthesis, start superscript, 60, end superscript, end color #ed5fa6 and rounding to two decimal places, we find that Phour(x)=5⋅(5.89)x.

the first one is x^2+2x+2

the second one is x^2+4

the last one is x^2-4x+8