A partially full glass of red grape juice is spilt and makes a stain on a carpet. Initially, the area of the stain is [math: 43] [math: cm2] cm^2 . In the next minute the area of the stain increases by [math: 4] [math: cm2] cm^2 and it continues to grow by [math: 80] % of the previous minute growth.The eventual area of the stain is:

Question

BlackTom AI · Accepted Answer

We start by restating the problem in our own words to ensure clarity: the stain begins with an area of 43 cm^2. In the first minute, the area increases by 4 cm^2. In every subsequent minute, the increase is 80% of the previous minute's increase, forming a decreasing geometric progression for the minute-by-minute increases. The eventual (total) area is the initial area plus the sum of all subsequent increases as time goes to infinity.

Option A: 80 cm^2. If the area eventually reached 80 cm^2, the total growth added to the initial 43 cm^2 would be 37 cm^2. Since the first minute adds only 4 cm^2, and later increases are 80% of the previous, the total sum of all increases from minute 1 onward is a finite geometric series with first term 4 and ratio 0.8. The sum of that infinite series is 4/(1-0.8) = 4/0.2 = 20 cm^2, not 37. So 80 cm^2 is too large.

Option B: 63 cm^2. The total eventual area would be the initial area plus the sum of all subsequent increases. The infinite sum of increases is S = a/(1-r) where a = 4 cm^2 and r = 0.8, giving S = 4/(1-0.8) = 20 cm^2. Therefore, eventual area = 43 + 20 = 63 cm^2. This matches the calculation from the geometric progression of minute-by-minute growth.

Option C: 60 cm^2. If the final area were 60 cm^2, the total added growth would be 17 cm^2. However, as shown, the infinite sum of increases is 20 cm^2, not 17, so 60 cm^2 cannot be correct given the stated growth pattern.

Option D: 67 cm^2. This would require the total increases to sum to 24 cm^2. But the actual infinite sum is 20 cm^2, so 67 cm^2 overestimates the final area based on the described growth process.

In summary, evaluating each option against the model of an initial 43 cm^2 area plus a first increase of 4 cm^2 followed by successive increases that are 80% of the previous one leads to an infinite increase sum of 20 cm^2 and a final area of 63 cm^2. The reasoning hinges on recognizing the increases form a geometric progression with first term 4 and ratio 0.8, and applying the formula for the sum of an infinite GP.

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希望你的学习变得更简单

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