Oil Spills The oil spilling from the ruptured hull of a grounded tanker spreads in all directions in caim waters. Suppose that the area polluted after t seconds is a circle of radius r feet, and the radius is increasing at the rate of 3ft/sec. (a) Find a function f giving the area (inf 2
) polluted in terms of r. f(r)= (b) Find a function g giving the radius (in ft ) of the polluted area in terms of t. g(f)= (c) Find a function h giving the area (in f 2
) polluted in terms of t. h(t)= (d) What is the size ( in ft 2
) of the polluted area 30sec after the hull was ruptured?
isaiah41201
(a) The area of a circle is given by the formula A = πr^2. Since the radius is increasing at a constant rate of 3 ft/sec, we can write the rate of change of the area with respect to time as dA/dt = (dA/dr)(dr/dt). In this case, (dA/dr) is equal to 2πr (since the derivative of r^2 with respect to r is 2r), and dr/dt is equal to 3. Therefore, dA/dt = 2πr(3) = 6πr. We can integrate this expression with respect to r to find the function f(r) that gives the area polluted in terms of r:
∫dA = ∫6πr dt
A = ∫6πr dt
A = 6π∫r dt
A = 6πrt + C
Since we are looking for the area at time zero when r = 0 (since the tanker is initially grounded), we can find the value of C:
A(t=0) = 6πr(0) + C
0 = 0 + C
C = 0
Therefore, the function f(r) giving the area polluted in terms of r is:
f(r) = 6πrt
(b) To find the function g giving the radius of the polluted area in terms of t, we can rearrange the equation from part (a):
f(r) = 6πrt
r = f(r)/(6πt)
Therefore, the function g giving the radius of the polluted area in terms of t is:
g(t) = f(r)/(6πt)
(c) To find the function h giving the area polluted in terms of t, we can substitute the expression for r from part (b) into the equation for f(r) in part (a):
h(t) = f(r) = 6πrt = 6π(f(r)/(6πt))(t)
h(t) = f(r) = f(r)
Therefore, the function h giving the area polluted in terms of t is:
h(t) = f(r) = 6πrt
(d) To find the size of the polluted area 30 seconds after the hull was ruptured, we can substitute t = 30 into the function h(t) from part (c):
h(t=30) = 6π(30)(30)
h(t=30) = 5400π
Therefore, the size of the polluted area 30 seconds after the hull was ruptured is 5400π ft^2.
(a) To find the function f giving the area polluted in terms of r, we need to use the formula for the area of a circle, which is given by A = πr^2. Since the area polluted is a circle with radius r, we can replace r with the increasing radius at a rate of 3 ft/sec, which is given by r = 3t. Substituting this into the formula for the area, we have:
f(r) = π(3t)^2
f(r) = π(9t^2)
f(r) = 9πt^2
(b) To find the function g giving the radius of the polluted area in terms of t, we can solve the equation r = 3t for t in terms of r:
r = 3t
t = r/3
Therefore, the function g is:
g(t) = r/3
(c) To find the function h giving the area polluted in terms of t, we can substitute the expression for r in terms of t into the function f:
h(t) = f(r)
h(t) = 9πt^2
(d) To find the size of the polluted area 30 seconds after the hull was ruptured, we can substitute t = 30 into the function h:
h(30) = 9π(30)^2
h(30) = 9π(900)
h(30) = 8100π
Therefore, the size of the polluted area 30 seconds after the hull was ruptured is 8100π square feet.