An oil rig breaks and begins to leak oil into the arctic ocean at an increasing rate of R(t) = 20t + 20 litres per day. At the instant of breakage (t=0), 100 litres of oil was spilled into the ocean via the emergency release valve. What is the equation V(t), the amount of oil (in litres) leaked into the ocean at time t days?
how do i use integrals?
So 20t +20 L/day is the rate of spilling oil
that is, dV/dt = 20t+20
V = 10t^2 + 20t + c
given: when t = 0, V = 100, then c = 100
V(t) = 10t^2 + 20t + 100
To find the amount of oil leaked into the ocean at time t, you can use an integral. Here's how you can proceed:
1. Start by finding the indefinite integral of the rate function R(t). In this case, the rate function is R(t) = 20t + 20.
∫(20t + 20) dt = 10t^2 + 20t + C
Note: C represents the constant of integration, which accounts for the fact that we don't have a specific starting point or initial condition.
2. Next, use the initial condition given in the problem. At t=0, 100 litres of oil were spilled into the ocean via the emergency release valve. This means that V(0) = 100.
3. Use this initial condition to find the specific value of the constant of integration. Substituting t=0 and V(0)=100 into the equation above:
100 = 10(0)^2 + 20(0) + C
100 = C
So, the specific equation for the amount of oil leaked into the ocean at time t days is:
V(t) = 10t^2 + 20t + 100
This equation will give you the amount of oil (in litres) leaked into the ocean as a function of time. You can substitute different values of t to find the amount of oil leaked at specific times.
To solve this problem, you can use integrals to find the equation V(t), which represents the amount of oil leaked into the ocean at time t.
First, let's analyze the situation. At the instant of the breakage (t=0), 100 liters of oil were spilled into the ocean. After that, the rate of oil leakage increases with time according to the function R(t) = 20t + 20 liters per day.
To find V(t), we need to integrate the rate function R(t) with respect to time. This will give us the accumulated amount of oil leaked into the ocean up to time t.
The integral of R(t) with respect to t will give us the antiderivative function of R(t), which we can call Q(t):
Q(t) = ∫ (20t + 20) dt
To find Q(t) using the integral, we can split the integral into two parts, one for each term:
Q(t) = ∫ 20t dt + ∫ 20 dt
Integrating each term:
Q(t) = 10t^2 + 20t + C
Where C is the constant of integration.
Since we know that at t=0, 100 liters of oil were spilled, we can substitute these values into the equation to solve for C:
Q(0) = 10(0)^2 + 20(0) + C = 100
This simplifies to:
C = 100
Now, we can write the equation V(t) for the amount of oil leaked at time t:
V(t) = Q(t) - Q(0) = 10t^2 + 20t + 100 - 100
Simplifying further, we get:
V(t) = 10t^2 + 20t
So, the equation V(t) represents the amount of oil leaked into the ocean at time t days.