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Question

# Seven days a year, Tiger Stadium becomes the _ fifth largest city in the state of Louisiana. Over 92,000 fans pack the stadium to watch the Tigers | play. After the game, if the fans leave at a rate of | 10% per minute, how long will it take before the | stadium is half empty? — a : :

Solution

Verified

Use the exponential decay function:

$y=a(1-r)^x$

where $a$ is the initial value and $r$ is the decay rate.



Substitute $a=92,000$ (fans) and $r=0.10$ (from 10% per minute) to find the equation:

$y=92,000 (1-0.10)^x$

$y=92,000 (0.90)^x$

Solve for $x$ when $y=\frac{1}{2}(92,000)=46,000$:

$46,000=92,000 (0.90)^x$

Divide both sides by 92,000:

$0.5= (0.90)^x$

Take the natural logarithm of both sides:

$\ln 0.5= \ln (0.90)^x$

Apply Power Property of logarithms:

$\ln 0.5= x\ln 0.90$

Divide both sides by $\ln 0.90$:

$\dfrac{\ln 0.5}{\ln 0.90}= x$

$x\approx \color{#c34632}6.6\text{ minutes}$

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