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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(1r)xy=a(1-r)^x

where aa is the initial value and rr is the decay rate.

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

y=92,000(10.10)xy=92,000 (1-0.10)^x

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

Solve for xx when y=12(92,000)=46,000y=\frac{1}{2}(92,000)=46,000:

46,000=92,000(0.90)x46,000=92,000 (0.90)^x

Divide both sides by 92,000:

0.5=(0.90)x0.5= (0.90)^x

Take the natural logarithm of both sides:

ln0.5=ln(0.90)x\ln 0.5= \ln (0.90)^x

Apply Power Property of logarithms:

ln0.5=xln0.90\ln 0.5= x\ln 0.90

Divide both sides by ln0.90\ln 0.90:

ln0.5ln0.90=x\dfrac{\ln 0.5}{\ln 0.90}= x

x6.6 minutesx\approx \color{#c34632}6.6\text{ minutes}

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