Question

Suppose a rumor is spreading that eating pickles will raise a​ person's IQ. Assume that 80 people have heard the rumor as of today and each day the number of people​ (past and​ present) who have heard it doubles. Let​ f(t) be the number of people​ (past and​ present) who have heard the rumor at t days since today.

a. Find an equation of f.
b. How many American (past and Present will have heard the rumor in 10 days from now)

Answer 1

To model the exponential growth of a rumor, we use the function f(t) = 80 × 2^t.

After 10 days, the number of people who have heard the rumor will be 81,920.

Step-by-step explanation:

The question pertains to an exponential growth problem where the number of people who have heard a rumor doubles every day. The goal is to find a mathematical function that represents this growth and to use it to calculate the number of people who will have heard the rumor after a certain period.

Part a:

The exponential growth function can be represented as f(t) = a × b^t, where a is the initial amount, b is the growth factor, and t is time in days. Given that the initial number of people who have heard the rumor is 80 and it doubles each day, the base b will be 2. Thus, the equation can be written as f(t) = 80 × 2^t.

Part b:

To find out how many people will have heard the rumor in 10 days, we substitute t with 10 in the equation we found in part a: f(10) = 80 × 2^10. Simplifying this, we get f(10) = 80 × 1024, which equals 81,920 people. Therefore, 81,920 people will have heard the rumor in 10 days.

Answer 2

[TeX]f(t)=80*2^{t-1} [/TeX]

f(10)=40960 people

Step-by-step explanation:

The number of people who have heard the rumour doubles every day.

If on the first day, 80 people have heard,

On the Second day, 160 persons would have heard.

This is an example of an exponential sequence whose growth ratio is 2.

The nth term of an exponential sequence is derived using the formula:

[TeX]U_n=ar^{n-1} [/TeX]

Where:

a=first term

r=growth ratio.

In this case, a=80, r=2

Therefore:

At any time t, the number who would have heard the rumor is:

[TeX]f(t)=80*2^{t-1} [/TeX]

(b)In 10 days,

[TeX]f(10)=80*2^{10-1} [/TeX]

[TeX]=80*2^{9} [/TeX]

=40960

In 20 days, 40960 persons would have heard the rumor.