Select the correct answer.
A tree has 99,400 leaves before autumn. Once autumn begins, the number of leaves on a tree decreases at a rate of $13 \%$ per day. After $t$ days of autumn, there are fewer than 12,000 leaves on the tree.
Which inequality represents this situation, and after how many days of autumn will the number of leaves on the tree be fewer than [tex]12,000[/tex]?
A. [tex]99,400(1.13)^t\ \textgreater \ 12,000 ; 15[/tex] days
B. [tex]99,400(0.87)^t\ \textless \ 12,000 ; 16[/tex] days
C. [tex]12,000(1.087)^t\ \textgreater \ 99,400 ; 18[/tex] days
D. [tex]12,000(0.913)^t\ \textless \ 99,400 ; 17[/tex] days
Answer (2)
The inequality representing the situation is 99 , 400 ( 0.87 ) t < 12 , 000 , and it will take approximately 16 days for the number of leaves to drop below 12,000. Thus, the correct answer is option B.
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The problem models the number of leaves decreasing by 13% each day, starting from 99,400 leaves.
Set up the inequality 99400 ( 0.87 ) t < 12000 to represent when the number of leaves is less than 12,000.
Solve for t by dividing both sides by 99400 and taking the natural logarithm, resulting in \frac{\ln(60/497)}{\ln(0.87)}"> t > l n ( 0.87 ) l n ( 60/497 ) .
Calculate t ≈ 15.3 , and round up to the next whole number, so the final answer is 16 .
Explanation
Understanding the Problem Let's analyze the problem. We start with 99,400 leaves, and the number decreases by 13% each day. This means we multiply by ( 1 − 0.13 ) = 0.87 each day. We want to find the inequality that represents when the number of leaves is less than 12,000, and the number of days it takes.
Setting up the Inequality The number of leaves after t days can be represented as 99400 ( 0.87 ) t . We want to find when this is less than 12,000, so the inequality is 99400 ( 0.87 ) t < 12000 .
Isolating the Exponential Term Now, let's solve for t . First, divide both sides of the inequality by 99400: ( 0.87 ) t < 99400 12000 = 994 120 = 497 60
Using Logarithms to Solve for t To solve for t , we can take the natural logarithm of both sides: t ln ( 0.87 ) < ln ( 497 60 ) Since ln ( 0.87 ) is negative, we need to reverse the inequality sign when we divide by it: \frac{\ln\left(\frac{60}{497}\right)}{\ln(0.87)}"> t > ln ( 0.87 ) ln ( 497 60 )
Calculating t Now, we calculate the value of t :
\frac{\ln\left(\frac{60}{497}\right)}{\ln(0.87)} \approx \frac{-2.126}{-0.139} \approx 15.3"> t > ln ( 0.87 ) ln ( 497 60 ) ≈ − 0.139 − 2.126 ≈ 15.3
Finding the Number of Days Since t must be an integer (number of days), we round up to the next whole number, which is 16. Therefore, after 16 days, the number of leaves will be fewer than 12,000.
Conclusion The correct inequality is 99 , 400 ( 0.87 ) t < 12 , 000 , and it will take 16 days for the number of leaves to be fewer than 12,000.
Examples
Imagine you're tracking the decay of a radioactive substance. The amount of the substance decreases by a certain percentage each day, similar to how the leaves decrease on the tree. You want to know when the amount of the substance falls below a safe level. This problem helps you determine how long you need to wait, using exponential decay and inequalities to model the situation and find the time it takes to reach the desired level.
