oil is leaking from a pipeline on the surface of a lake and forms an oil slick whose volume increases at a constant rate of 2000 cubic centimeters per minute. The oil slick takes the form of a right circular cylinder with both its radius r and height changing h is given by V= pie times r to the second power times h)
b) a recovery device arrives on the scene and begins removing oil. The rate at which oil is removed is R(t)=400 times square root of t cubics per min, where t is the time in min since the device began working.oil continues to leak at the rate of 2000 cubic cm per min. Find t when oil slick reaches its max volume.
c) By the time the recovery device began removing oil 60000 cubic cm of oil had already leaked. write but not evauate and expression involving an integral that gives the volume of oil at the time found in part (b)
help please start them
V = pi r^2 h
dV/dt = 2000 - 400 t^.5
b) when is dV/dt = 0?
400 t^.5 = 2000
sqrt t = 5
t = 25 min
c) V at t=0 = 60,000
dV/dt = 2000 -400 t^.5
V = 60,000 + integral from t = 0 to t = 25 of (2000 - 400 t^.5) dt
To solve part (b), we need to find the time when the oil slick reaches its maximum volume.
To begin, let's establish the equation that relates the volume of the oil slick to its height and radius.
We are given the equation: V = πr^2h, where V is the volume in cubic centimeters, r is the radius in centimeters, and h is the height in centimeters.
Since the oil slick is in the form of a right circular cylinder, we can express the height in terms of time, h(t), as h(t) = h0 + rt, where h0 is the initial height of the oil slick and r is the rate at which the height changes.
Similarly, we can express the radius in terms of time, r(t), as r(t) = r0 + st, where r0 is the initial radius of the oil slick and s is the rate at which the radius changes.
Substituting the expressions for h(t) and r(t) into the equation for the volume, we get:
V(t) = π(r0 + st)^2(h0 + rt)
Now, we know that the volume of the oil slick is increasing at a constant rate of 2000 cubic centimeters per minute. So, we have:
dV/dt = 2000
To find the maximum volume, we need to find the time when the rate of change of volume is zero. Therefore, we differentiate V(t) with respect to t and set it equal to zero:
dV/dt = 2π(r0 + st)(s(h0 + rt) + r0 + st) = 0
Now, let's solve this equation:
2π(r0 + st)(s(h0 + rt) + r0 + st) = 0
Simplifying this equation and dividing both sides by 2π, we get:
(r0 + st)(s(h0 + rt) + r0 + st) = 0
Expanding and rearranging, we have:
s^2h0t + r0^2s + 2r0st + s^2rt^2 = 0
This equation is a quadratic equation in terms of t. By solving this equation, we can find the time when the oil slick reaches its maximum volume.
I will stop here for now. Let me know if you need further assistance or want me to continue with part (c).
To solve parts (b) and (c) of the problem, we need to understand the relationship between the oil slick's volume, the oil leak rate, and the rate at which oil is being removed by the recovery device. Let's break down the problem step by step:
Step 1: Understanding the volume of the oil slick
The volume of the oil slick can be calculated using the formula: V = π * r^2 * h, where V is the volume, r is the radius of the circular cylinder, and h is its height. According to the problem statement, the volume of the oil slick increases at a constant rate of 2000 cubic centimeters per minute.
Step 2: Understanding the rate of oil removal
The rate at which oil is being removed by the recovery device is given by the function R(t) = 400 * √(t), where t is the time in minutes since the device began working. This function represents the rate of oil removal in cubic centimeters per minute.
Step 3: Determining the time when the oil slick reaches maximum volume (part b)
To find the time when the oil slick reaches its maximum volume, we need to find the point where the rate of oil removal equals the leak rate. In other words:
Leak rate = Oil removal rate
Since the leak rate is constant at 2000 cubic centimeters per minute, we can set it equal to the derivative of the volume function V with respect to time (dV/dt). Then, we solve this equation to find the value of t.
Step 4: Expressing the volume of oil leaked using an integral (part c)
To express the volume of oil leaked by the time the recovery device began working (60000 cubic centimeters), we can write an integral with a variable of integration from time (t = 0) to the time found in part (b).
Now that we've understood the problem and the approach, let's proceed to solve parts (b) and (c) step by step.