There is a sudden growth of algae in a lake that kills other life in the water such that y tons of algae will be present after t weeks. The rate of growth of the algae is y′=ty+t and the initial amount of algae is 7 tons. How long will it take the initial amount of algae to grow to 77 tons? (Round your answer to three decimal places.)
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There is a sudden growth of algae in a lake that kills other life in the water such that y tons of algae will be present after t weeks. The rate of growth of the algae is y′=ty+t and the initial amount of algae is 7 tons. How long will it take the initial amount of algae to grow to 77 tons? (Round your answer to three decimal places.)
To find out how long it will take for the initial amount of algae to grow to 77 tons, we need to solve the equation y² = ty + t, where y represents the quantity of algae and t represents time in weeks.
Given that the initial amount of algae is 7 tons, we can substitute y = 7 into the equation:
7² = 7t + t²
49 = 7t + t²
To find when the initial amount will grow to 77 tons, we can substitute y = 77 into the equation and solve for t:
77² = 77t + t²
5,929 = 77t + t²
Now we have a quadratic equation, so we can rearrange it to get one side equal to zero:
t² + 77t - 5,929 = 0
We can use the quadratic formula to find the values of t:
t = (-b ± sqrt(b² - 4ac)) / 2a
In this case, a = 1, b = 77, and c = -5929. Plugging these values into the quadratic formula gives us:
t = (-77 ± sqrt(77² - 4(1)(-5929))) / 2(1)
Simplifying the equation further gives us:
t = (-77 ± sqrt(5,929 + 23,716)) / 2
t = (-77 ± sqrt(29,645)) / 2
t = (-77 ± 171.967) / 2
There are two possible solutions:
t₁ = (-77 + 171.967) / 2 = 47.983
t₂ = (-77 - 171.967) / 2 = -124.967 (not meaningful in this context)
Therefore, it will take approximately 48 weeks for the initial amount of algae to grow to 77 tons.
To find out how long it will take for the initial amount of algae to grow to 77 tons, we need to solve for the time (t) in the given equation:
y² = ty + t
First, let's substitute the initial amount of algae (7 tons) into the equation:
(7)² = 7t + t
49 = 8t
Now, let's solve for t by dividing both sides of the equation by 8:
t = 49/8
t = 6.125
So, it will take approximately 6.125 weeks for the initial amount of algae to grow to 77 tons.