suppose that oil spilled from a tanker and spreads in a circular pattern whose radius increases at a constant rate of .5 feet per second.
-find a function f that models the radius of the oil spill as a function of time r=f(t) with t measured in seconds. How long before the radius reaches 60 feet? how large is the radius after 1 minute? after 1 hour?
-find a function g that models the area (sq. feet) of the oil spill as a function of the radius A=g(r) Calcuate the area (sq. ft.) covered by oil when the radius reaches 60 feet.
-find the composite function h=g*f what does this fuction represent? evaluate h(30) = (g*f)(30) and explain what this represents include units.
-calcuate the area (Sq ft) covered by oil 30 seconds after the taker ruptured 1 minute after the tanker ruptured and 1 hour after the taker ruptured
-calculate the average rate of change of the amout of oil spilled sq feet per second from t=0 to t=30 seconds and from t=30 seconds to t=60 seconds is the area of the oil spill chaging at a cosntant rate with time? what does this imply? what does the inverse function h^-1 represent? Evaluate h^-1(30) and explain what this represesnts include unites
To find the function that models the radius of the oil spill as a function of time, we need to determine the relationship between the radius and time. We know that the radius increases at a constant rate of 0.5 feet per second.
Let's start by considering the initial conditions. At time t = 0 seconds, the radius is 0 feet. As time progresses, the radius will increase at a rate of 0.5 feet per second. Therefore, we can write the function as:
r = f(t) = 0.5t
To calculate how long it takes for the radius to reach 60 feet, we can substitute r = 60 into the equation and solve for t:
60 = 0.5t
t = 120 seconds
So, it will take 120 seconds (or 2 minutes) for the radius to reach 60 feet.
To find the radius after 1 minute (or 60 seconds), we substitute t = 60 into the equation:
r = 0.5(60) = 30 feet
The radius after 1 minute is 30 feet.
To find the radius after 1 hour (or 3600 seconds), we substitute t = 3600 into the equation:
r = 0.5(3600) = 1800 feet
The radius after 1 hour is 1800 feet.
Now let's find a function g that models the area of the oil spill as a function of the radius, A = g(r). The area of a circle is given by the formula A = πr^2.
So, we substitute r into this formula:
A = πr^2
A = π(0.5t)^2
A = π(0.25t^2)
To calculate the area covered by oil when the radius reaches 60 feet, we substitute r = 60 into the equation:
A = π(0.25(60)^2)
A = π(0.25(3600))
A = π(900)
A = 900π square feet
The area covered by oil when the radius reaches 60 feet is 900π square feet.
Next, we are asked to find the composite function h = g * f, which represents the area covered by oil as a function of time.
h(t) = g(f(t)) = g(0.5t)
To evaluate h(30), we substitute t = 30 into the equation:
h(30) = g(0.5(30))
h(30) = g(15)
The units of h(30) will be in square feet as it represents the area.
To calculate the area covered by oil 30 seconds after the tanker ruptured, we substitute t = 30 into the equation for h:
h(30) = g(0.5(30))
h(30) = g(15)
To calculate the area covered by oil 1 minute after the tanker ruptured, we substitute t = 60 into the equation for h:
h(60) = g(0.5(60))
h(60) = g(30)
To calculate the area covered by oil 1 hour after the tanker ruptured, we substitute t = 3600 into the equation for h:
h(3600) = g(0.5(3600))
h(3600) = g(1800)
Next, we are asked to calculate the average rate of change of the amount of oil spilled in square feet per second from t = 0 to t = 30 seconds, and from t = 30 seconds to t = 60 seconds.
To find the average rate of change, we find the difference in the area covered by oil and divide it by the difference in time:
Average rate of change from t = 0 to 30 seconds:
Change in area = h(30) - h(0)
Change in time = 30 - 0
Average rate of change = (h(30) - h(0)) / (30 - 0)
To find the average rate of change from t = 30 seconds to 60 seconds, we substitute the corresponding values into the formula:
Average rate of change from t = 30 to 60 seconds:
Change in area = h(60) - h(30)
Change in time = 60 - 30
Average rate of change = (h(60) - h(30)) / (60 - 30)
If the value of the average rate of change is constant, it implies that the area of the oil spill is changing at a constant rate with time.
The inverse function h^(-1) represents the time it takes for a certain area to be covered by oil.
To evaluate h^(-1)(30), we calculate the time it takes for an area of 30 square feet to be covered by oil. We substitute A = 30 into the equation for h:
30 = g(f(t))
We solve for t to find the time:
t = h^(-1)(30)
The units of t will be in seconds, as it represents the time.