A radioactive substance is leaking from an underground storage tank. It spread 1000 meters from the source during the first year, 800 meters the second year, 640 meters the third year ( hint think concentric circles). a) If the pattern continues, how far from the center will it have spread by the end of the year? b) Using formulas to prove your answer, determine whether the radioactive substance ever reaches a river 6000 meters from the source
The radius grows as a geometric series
1000+800+640+...
where
a = 1000
r = 0.8
Since 1000/(1-0.8) = 5000, the sum will never get to 6000
To solve this problem, we can observe that the distance the radioactive substance spreads forms a series of concentric circles. Each year, the substance spreads a certain percentage of the previous year's distance.
Let's break down the problem into two parts:
a) How far from the center will the substance have spread by the end of the year?
To find the answer, we need to consider the pattern formed by the substance's spreading. Each year, it spreads a certain percentage of the previous year's distance.
Observing the given pattern, we can see that the substance spreads by 80% of the previous year's distance each year. This means that each year, the new radius is 80% of the previous year's radius.
Starting with a radius of 1000 meters, we can calculate the radius at the end of each year:
Year 1: 1000 meters * 0.8 = 800 meters
Year 2: 800 meters * 0.8 = 640 meters
Year 3: 640 meters * 0.8 = 512 meters
By the end of the year, the substance will have spread approximately 512 meters from the center.
b) Determining whether the substance reaches a river 6000 meters from the source using formulas:
To determine whether the substance reaches a river 6000 meters away, we need to compare the distance the substance spreads each year with the distance to the river.
Let's denote "r" as the radius of the spreading substance and "d" as the distance to the river. If the radius is equal to or greater than the distance to the river, then the substance will reach the river.
We can use the following formula to calculate the radius of the substance at the end of the year:
r = Initial radius * (Rate of spreading)^n
Using this formula, where the initial radius is 1000 meters and the rate of spreading is 80% (0.8), we can substitute "n" as the number of years.
r = 1000 * (0.8)^n
To determine whether the substance reaches the river 6000 meters away, we can set up the following equation:
1000 * (0.8)^n >= 6000
We can now solve this equation to find the value of "n":
(0.8)^n >= 6
Taking the logarithm of both sides, we get:
n * log(0.8) >= log(6)
n >= log(6) / log(0.8)
Calculating this expression, we find that n >= 10.448 (approximately).
Since "n" represents the number of years, we can conclude that the substance will reach or surpass the river after approximately 11 years.
Therefore, using formulas, we can determine that the radioactive substance will spread around 512 meters from the center by the end of the year, and it will reach the river after approximately 11 years.