Question

Jamie is starting a new marketing strategy that causes her number of clients to grow exponentially. Jamie starts with 50 clients, and each month, her number of clients doubles. The expression 50⋅2^m can be used to model the expression. Jamie found that after just 6 months, she had reached 3,200 clients. How did Jamie reach this solution?
A) Jamie substituted m=6 into the expression 50 ⋅ 2 ⋅ 6 = 600.
B) Jamie substituted m=6 into the expression 6 ⋅ 2^50 = 3200.
C) Jamie substituted m = 6 into the expression 50 ⋅ 2^6 = 1800.
D) Jamie substituted m = 6 into the expression 50 ⋅ 2^6 = 3200.

Answer 1

To determine how Jamie reached her solution, let's analyze the given expression and options.

The expression given is \( 50 \cdot 2^m \), where \( m \) represents the number of months. Jamie starts with 50 clients, and her number of clients doubles each month.

To find the number of clients after 6 months, we set \( m = 6 \):

\[
\text{Number of clients} = 50 \cdot 2^6
\]

Now, we calculate \( 2^6 \):

\[
2^6 = 64
\]

Next, we multiply this by 50:

\[
50 \cdot 64 = 3200
\]

So, after substituting \( m = 6 \) into \( 50 \cdot 2^m \), Jamie indeed finds that she has 3200 clients after 6 months.

Now, let's review the provided options:

A) Jamie substituted \( m=6 \) into the expression \( 50 \cdot 2 \cdot 6 = 600 \).
- This is incorrect; this does not use the correct formula.

B) Jamie substituted \( m=6 \) into the expression \( 6 \cdot 2^{50} = 3200 \).
- This is incorrect; it uses the wrong formula (and \( 2^{50} \) is much larger than required).

C) Jamie substituted \( m=6 \) into the expression \( 50 \cdot 2^6 = 1800 \).
- This is incorrect; \( 50 \cdot 2^6 \) equals 3200, not 1800.

D) Jamie substituted \( m=6 \) into the expression \( 50 \cdot 2^6 = 3200 \).
- This is correct, as it matches our calculations.

The correct answer is:

**D) Jamie substituted m = 6 into the expression 50 ⋅ 2^6 = 3200.**