Question

Ryan has the chickenpox! He was told that the number of pockmarks on his body would grow exponentially until his body overcomes the illness. He found that he had 60 pockmarks on November 1, and by November 3 the number had grown to 135. To find out when the first pockmark appeared, he will need to find the exponential function that will model the number of pockmarks based on the day. a. Ryan decides to find the exponential function that passes through the points (3, 135) and (1, 60). Use these points to write the equation of his function of the form f(x)=abˣ b. According to your model, what day did Ryan get his first chickenpox pockmark?

Answer 1

The answer is 22 of October.

Explanation:

The final exponential model must be of the form
`f(x) = ab^(x)`. Where f(x) is the number of pockmarks, x is the day number and a is a multiplicative constant and b is the base.

we plug in the coordinates (3,135), we get,

`135 = ab^(3)`

call it equation 1. Now we plug in the second coordinate (1,60) to obtain.

`60 = ab^(1)`

call it equation 2. If we divide equation 1 by 2 to eliminate a we get,

`135/60 = b^(2)`

`b = 3/2`

plugging this back into equation 2 , that is 60=a*(3/2), we obtain,

`a = 40`

Now that we have the values of both a and b ,we can plug them back into the main equation obtaining,

`f(x) = 40 * (3/2)^(x)`

we have obtained the equation of Ryans exponential function. Now to answer the last part of the question that is what day did Ryan get his first chicken pox pockmark , we just plug in 1 in place of f(x) to obtain the corresponding x.

`1 = 40*(3/2)^(x)`

this can be rewritten in the logarithmic form as ,

`x = log(3/2)(1/40) = -9.098`

Since the magnitude of the number is greater than 9, we will have to count that as only just spilling to the 10th day. Which only means that 10 days less than the first day of November and since the month of October has 31 days , we get that 10 days before the 1st of November Ryan got his first pockmark that is 22nd of October.