There's a question I've been having difficulty solving, and I would really appreciate it if you can explain to me how I can achieve the answer.
question:an above ground swimming pool in the shape of a cylinder, with diameter 5m, is filled at a constant rate to a depth of 1m. It takes 4h to fill the pool with a hose. Create an equation of a mathematical model for volume as a function of time.
the surface of the water is a circle of diameter 5m, so its area is
25π/4 m^2
Since it takes 4 hrs to fill the 1m, pool, the water comes in at a rate of
25π/16 m^3/hr
So, after t hours, the volume of water in the pool is
v(t) = 25π/16 t m^3
To create a mathematical model for volume as a function of time, we need to consider the formula for the volume of a cylinder.
The formula for the volume of a cylinder is given by V = πr^2h, where V is the volume, r is the radius (half the diameter), and h is the height.
In this problem, the diameter of the swimming pool is given as 5m. So, the radius (r) would be half of the diameter, which is 2.5m.
Now, let's focus on finding the volume of the cylinder as a function of time.
The pool is being filled with a hose at a constant rate, which means the rate of change of the volume of water in the pool with respect to time is constant.
Let's assume the volume of water in the pool at time t is denoted by V(t). We know that when t = 0, the volume of water in the pool is 0. As time progresses and the pool gets filled, the volume of water increases.
Now, given that it takes 4 hours to fill the pool, we can calculate the rate of change of the volume of water, also known as the filling rate. It is given by the total volume of water (V) divided by the time taken (t).
So, the filling rate (r) can be calculated as r = V / t.
In our case, the volume of water in the pool is the volume of the cylinder, V = πr^2h, where r = 2.5m (radius) and h = 1m (depth).
Now, let's substitute the values and calculate the filling rate (r) for the given values.
r = V / t
r = (π * 2.5^2 * 1) / 4
Simplifying this equation gives us the filling rate as r = 3.125π m^3/h.
Now, we have the filling rate (r) in terms of the volume of water per hour. To get the equation of the mathematical model for volume as a function of time, we can rearrange the filling rate equation:
V = r * t
Substituting the filling rate (r) we calculated earlier, the equation becomes:
V = 3.125π * t
Therefore, the mathematical model for volume as a function of time is V(t) = 3.125πt, where V(t) represents the volume of water in the pool at time t.