The spread of oil leaking from a tanker after t hours is in the shape of a circle with radius
r(t) = 200 square root t+3, - 600 (in feet).
A. Find the time it takes for the leak to spread at a radius of 1800ft.
B. Find the area A of the oil leak as a function of time t.
A.
substitute r = 1800 to the equation:
1800 = 200*sqrt(t+3) - 600
2400 = 200*sqrt(t+3)
12 = sqrt(t+3)
144 = t+3
t = 141 hours
B.
recall that area of circle is just A = pi*r^2. thus we square both sides and multiply by pi:
r(t) = 200*sqrt(t+3) - 600
r^2 = 40000(t+3) - 240000*sqrt(t+3) - 360000
pi*r^2 = A = pi*[40000(t+3) - 240000*sqrt(t+3) - 360000]
we can further simplify this as,
A = pi*[40000t + 120000 - 240000*sqrt(t+3) - 360000]
A = pi*[40000t - 240000*sqrt(t+3) - 240000)
hope this helps~ :)
A. To find the time it takes for the leak to spread at a radius of 1800ft, we need to solve the equation r(t) = 1800.
The equation r(t) = 200 √(t+3) - 600 is given, and we want to find the value of t when r(t) = 1800.
200 √(t+3) - 600 = 1800
Add 600 to both sides of the equation:
200 √(t+3) = 2400
Divide both sides by 200:
√(t+3) = 12
Square both sides of the equation:
t+3 = 144
Subtract 3 from both sides of the equation:
t = 141
Therefore, it takes approximately 141 hours for the leak to spread to a radius of 1800ft.
B. To find the area A of the oil leak as a function of time t, we need to use the formula for the area of a circle:
A = π * r(t)^2
Substitute the given expression for r(t):
A = π * (200 √(t+3) - 600)^2
Simplify the expression by squaring the quantity inside the parentheses:
A = π * (200^2 * (t+3) - 2 * 200 * 600 * √(t+3) + 600^2)
Simplify further by expanding and combining like terms:
A = π * (40000t + 120000 - 240000 * √(t+3) + 360000)
A = π * (40000t + 480000 - 240000 * √(t+3))
Therefore, the area A of the oil leak as a function of time t is A(t) = π * (40000t + 480000 - 240000 * √(t+3)).