To find how many quarter years [tex]\( q \)[/tex] it takes for the number of active players to decrease by 10%, we need to solve the equation given the function:
[tex]\[ A(q) = 86 \left( 0.9 \right)^{\frac{q}{4}} \][/tex]
Here are the step-by-step instructions to solve the problem:
1. Initial Number of Players:
The number of active players initially when [tex]\( q = 0 \)[/tex] is [tex]\( A(0) \)[/tex].
[tex]\[ A(0) = 86 \][/tex]
2. Target Number of Players:
A decrease of 10% means the new number of active players should be 90% of the initial value:
[tex]\[ A_{\text{target}} = 86 \times 0.9 = 77.4 \][/tex]
3. Form the Equation:
We set the function equal to the target number of active players:
[tex]\[ 86 \left( 0.9 \right)^{\frac{q}{4}} = 77.4 \][/tex]
4. Isolate the Exponential:
Divide both sides by 86 to isolate the exponential term:
[tex]\[ \left( 0.9 \right)^{\frac{q}{4}} = \frac{77.4}{86} \][/tex]
[tex]\[ \left( 0.9 \right)^{\frac{q}{4}} = 0.9 \][/tex]
5. Solve for [tex]\( \frac{q}{4} \)[/tex]:
Both sides have the same base (0.9), so set the exponents equal to each other:
[tex]\[ \frac{q}{4} = 1 \][/tex]
6. Solve for [tex]\( q \)[/tex]:
Multiply both sides by 4 to isolate [tex]\( q \)[/tex]:
[tex]\[ q = 4 \][/tex]
Thus, it takes [tex]\( \boxed{4} \)[/tex] quarter years for the number of active players to decrease by 10%.
