Sam N.
asked • 08/03/20Calculus: Related rates A tank leaks 0.13 m^3/hr of oil into a lake
A tank leaks 0.13 m^3/hr of oil into a lake. The oil forms a semicircular disk with a thickness of 10^-6 meters. How rapidly is the radius of the disk increasing 3 hours after the tank begins leaking? The radius of the disk is increasing by ______ m/hr
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1 Expert Answer
Mike D. answered • 08/03/20
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Sam
If the radius of the disk is r metres, then the volume of the disk will be V = 10^-6 πr2
If t is measured in hours we know dV/dt = 0.13
dV/dt is also 10^-6 . π . 2 r . dr/dt
You want dr/dt when t=3. Well you nearly have enough information. If you knew r, you could set 0.13 = the above expression and solve for dr/dt.
You need to find r, when t=3. Well obviously at t=3, V = 0.13 X 3 = 0.39. Using this and the volume expression you can find r.
Mike
Sam N.
Thank you! ANSWER: V = (1/2)πr^2*t Assume that the thickness t = 10^(-6) metres remains constant semicircular disk creates a half cylinder volume V = (π/2)r^2*10^-6 ............(1) dV/dr = π*10^-6 * r dV/dt = dV/dr * dr/dt dr/dt = (dV/dt)/(dV/dr) = [0.13/(πr)]*10^6 We need to know r after 3 hours 3*0.13 = (π/2)r^2*10^-6 πr^2 = 0.78 *10^6 r ~ 498.3 m πr ~ 1565.39 dr/dt = [0.13/(1565.39)]*10^6 ~ 83.05 metres per hour
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08/03/20
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Sam N.
V = (1/2)πr^2*t Assume that the thickness t = 10^(-6) metres remains constant semicircular disk creates a half cylinder volume V = (π/2)r^2*10^-6 ............(1) dV/dr = π*10^-6 * r dV/dt = dV/dr * dr/dt dr/dt = (dV/dt)/(dV/dr) = [0.13/(πr)]*10^6 We need to know r after 3 hours 3*0.13 = (π/2)r^2*10^-6 πr^2 = 0.78 *10^6 r ~ 498.3 m πr ~ 1565.39 dr/dt = [0.13/(1565.39)]*10^6 ~ 83.05 metres per hour08/03/20