To solve this problem, we can set up an equation to represent the situation. Let x be the original number of fish in the lake.
After the first increase of 60%, the number of fish in the lake becomes 1.60x = 1.6x. 180 fish are removed, leaving 1.6x - 180 fish in the lake.
After the second increase of 60%, the number of fish in the lake becomes 1.60(1.6x - 180) = 1.6(1.6x - 180) = 1.6(1.6x) - 1.6(180) = 1.6 * 1.6x - 1.6 * 180 = 1.6^2x - 288.
For the number of fish in the lake to remain constant, we need to find x such that 1.6^nx - 180n = x, where n is the number of times the increase and removal process is repeated.
Now, let's substitute in the values of the multiple increase and removal process:
1.6^2x - 180*2 = x
1.6^3x - 180*3 = x
1.6^nx - 180n = x
We are looking for a value of x for which the above equations hold true. Let's find the specific value:
1.6^nx = 180n
x = (180n) / (1.6^n)
We want to find a value of x where these equations hold true, regardless of n. Let's look at the answer choices and see which one gives us a constant value for x:
A. 180180 = (180*1)/(1.6^1) = 180/1.6 = 112.5
B. 220220 = (180*2)/(1.6^2) = 360/2.56 = 140.625
C. 260260 = (180*3)/(1.6^3) = 540/4.096 = 131.8359375
D. 300300 = (180*4)/(1.6^4) = 720/6.5536 = 109.765625
E. 340340 = (180*5)/(1.6^5) = 900/10.48576 = 85.9375
From the calculations above, we can see that the only possible constant value for x is 111.5. Therefore, the correct answer is:
A. 180180
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Question
A city has just passed an ordinance to control the number of fish in a lake. When the number of fish in a lake increases by 60%, 180 fish must immediately be removed from the lake. When this new number of fish increases by 60%, 180 fish must immediately be removed from the lake again, and so on. If the ordinance is followed, which number will result in the number of fish in the lake remaining constant?
Responses
A 180180
B 220220
C 260260
D 300300
E 340340
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