The correct function that models the change in the number of coffee shops over time, given that the number of shops is reduced to one-half each year, is:
[tex]\[ d(t) = 35,000 \cdot \left(\frac{1}{2}\right)^t \][/tex]
Here’s how we derive and understand this function, step by step:
1. Initial Number of Stores:
The chain starts with 35,000 stores worldwide.
2. Reduction Factor:
Each year, the number of stores is reduced to one-half of the previous year's number.
3. Function Representation:
To find the number of stores remaining after [tex]\( t \)[/tex] years, we will use an exponential decay function. The general form of an exponential decay function is:
[tex]\[ N(t) = N_0 \times r^t \][/tex]
where:
- [tex]\( N(t) \)[/tex] is the number of stores remaining after [tex]\( t \)[/tex] years.
- [tex]\( N_0 \)[/tex] is the initial number of stores.
- [tex]\( r \)[/tex] is the decay rate (in this case, the reduction factor).
- [tex]\( t \)[/tex] is the time in years.
4. Applying the Values:
Given:
- Initial number of stores [tex]\( N_0 = 35,000 \)[/tex]
- Reduction factor [tex]\( r = \frac{1}{2} \)[/tex]
5. Modeling the Function:
Substituting these values into the exponential decay formula:
[tex]\[ d(t) = 35,000 \cdot \left(\frac{1}{2}\right)^t \][/tex]
Now, let's use this function to calculate the number of stores remaining after 3 years:
[tex]\[ d(t) = 35,000 \cdot \left(\frac{1}{2}\right)^3 \][/tex]
Breaking it down:
- Step 1: Calculate [tex]\(\left(\frac{1}{2}\right)^3\)[/tex]:
[tex]\[ \left(\frac{1}{2}\right)^3 = \frac{1}{2} \times \frac{1}{2} \times \frac{1}{2} = \frac{1}{8} \][/tex]
- Step 2: Multiply the initial number of stores by this value:
[tex]\[ d(3) = 35,000 \cdot \frac{1}{8} = 35,000 \cdot 0.125 \][/tex]
- Step 3: Perform the multiplication:
[tex]\[ 35,000 \times 0.125 = 4,375 \][/tex]
Therefore, after 3 years, the number of remaining stores will be:
[tex]\[ 4,375 \][/tex]
So, the function [tex]\( d(t) = 35,000 \cdot \left(\frac{1}{2}\right)^t \)[/tex] correctly models the number of remaining stores, and after 3 years, there will be 4,375 stores left.
