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Venera sent a chain letter to her friends, asking them to forward the letter to more friends.

The relationship between the elapsed time [tex]\( t \)[/tex], in months, since Venera sent the letter, and the number of people, [tex]\( P(t) \)[/tex], who receive the email is modeled by the following function:
[tex]\[ P(t) = 2^{3t+7} \][/tex]

Complete the following sentence about the monthly rate of change in the number of people who receive the email. Round your answer to two decimal places.

Every month, the number of people who receive the email is multiplied by a factor of [tex]\(\square\)[/tex].



Answer :

GinnyAnswer
Given the function [tex]\( P(t) = 2^{3t + 7} \)[/tex], we aim to determine the rate at which the number of people receiving the email changes each month.

1. Identify the increase in [tex]\( t \)[/tex] over one month:

When [tex]\( t \)[/tex] increases by 1 month, we replace [tex]\( t \)[/tex] with [tex]\( t + 1 \)[/tex].

The function expressing the number of people becomes:
[tex]\[ P(t + 1) = 2^{3(t + 1) + 7} \][/tex]

2. Simplify the exponent for [tex]\( P(t + 1) \)[/tex]:
[tex]\[ P(t + 1) = 2^{3t + 3 + 7} = 2^{3t + 10} \][/tex]

3. Express the factor of change by comparing [tex]\( P(t + 1) \)[/tex] and [tex]\( P(t) \)[/tex]:

The ratio between the number of people in [tex]\( t + 1 \)[/tex] months and [tex]\( t \)[/tex] months gives the factor by which the number of people is multiplied each month:
[tex]\[ \text{Factor} = \frac{P(t + 1)}{P(t)} = \frac{2^{3t + 10}}{2^{3t + 7}} \][/tex]

4. Simplify using properties of exponents:

When dividing powers with the same base, subtract the exponents:
[tex]\[ \frac{2^{3t + 10}}{2^{3t + 7}} = 2^{(3t + 10) - (3t + 7)} = 2^{3} \][/tex]

Thus, the factor of change each month is:
[tex]\[ 2^3 = 8 \][/tex]

Therefore, every month, the number of people who receive the email is multiplied by a factor of [tex]\( 8 \)[/tex].