Question

A company owns 2 locations and wants to compare their weekly profits. The function f(x)=4^x-2 gives locations A profits in thousands of dollars after x weeks
Location B function g(x)=x^2 profits in thousands of water x weeks. When will each location have a profit of $2,000
Options
Location A in 1 week and location B in 1.4 weeks
Location A 5.5 weeks and location B 44.7 weeks
Location A 1.4 weeks and location B 1 week
Both in less than a week

Answer 1

To determine when each location will have a profit of $2,000, we need to find the values for \( x \) at which the profit functions \( f(x) \) and \( g(x) \) equal 2 (since the profits are given in thousands of dollars).

1. **For Location A:**
\[ f(x) = 4^x - 2 \]
Set this equal to 2:
\[
4^x - 2 = 2
\]
\[
4^x = 4
\]
\[
4^x = 4^1
\]
This implies:
\[
x = 1
\]

2. **For Location B:**
\[ g(x) = x^2 \]
Set this equal to 2:
\[
x^2 = 2
\]
\[
x = \sqrt{2} \approx 1.414
\]
(We only consider the positive solution since time cannot be negative).

Now, let's summarize the results:
- Location A will reach a profit of $2,000 in **1 week**.
- Location B will reach a profit of $2,000 in approximately **1.4 weeks**.

Therefore, the correct answer from the options provided is:

**Location A in 1 week and location B in 1.4 weeks**.