The number f water hyacinth plants growing on a lake increases exponentially with time. On Tuesday the number of plants is 150. That Friday the number has risen to 240. If it takes 3400 plants to completely cover the lake, will the lake be completely covered by 20 days after Tuesday? Justify your answer.
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To determine if the lake will be completely covered by 20 days after Tuesday, we need to find the growth rate of the water hyacinth plants per day. Let's set up an exponential growth model based on the information given:
N(t) = N0 * e^(kt)
Where:
N(t) is the number of plants at time t
N0 is the initial number of plants
e is the base of the natural logarithm (approximately 2.71828)
k is the growth rate
t is the time in days
We know that on Tuesday (t=0), the number of plants is 150 (N0 = 150). That Friday (t=4), the number of plants is 240. We can use this information to find k.
240 = 150 * e^(4k)
Divide both sides by 150:
e^(4k) = 240/150
e^(4k) = 8/5
Take the natural logarithm of both sides:
ln(e^(4k)) = ln(8/5)
4k = ln(8/5)
Divide both sides by 4:
k = ln(8/5) / 4
Now we can plug in the value of k into our growth model to find the number of plants after 20 days:
N(t) = 150 * e^(kt)
N(20) = 150 * e^((ln(8/5) / 4) * 20)
Using a calculator, we can evaluate N(20) to find the number of plants after 20 days. If the result is greater than or equal to 3400, then the lake will be completely covered. Otherwise, it won't be.
Based on the calculations, if the number of water hyacinth plants on the lake follows an exponential growth pattern, it is unlikely that the lake will be completely covered by 20 days after Tuesday since the number of plants will not reach 3400.