The radius of a right circular cylinder is given by t + 9
and its height is (1/9)*(t^0.5)
where t is time in seconds and the dimensions are in inches. Find the rate of change of the volume with respect to time.
units = in^3/s
To find the rate of change of the volume with respect to time, we need to differentiate the volume formula with respect to time.
The volume of a right circular cylinder is given by V = πr^2h, where r is the radius and h is the height.
From the given information, we have:
r = t + 9
h = (1/9) * t^0.5
Substituting these values into the volume formula, we get:
V = π(t + 9)^2 * (1/9)*t^0.5
To find dV/dt, we differentiate the volume formula with respect to t:
dV/dt = d/dt [π(t + 9)^2 * (1/9)*t^0.5]
Using the product rule and chain rule, we can differentiate this expression step-by-step.
Let's start by differentiating the first term: π(t + 9)^2
d/dt [π(t + 9)^2] = 2π(t + 9) * (1)
Next, let's differentiate the second term: (1/9)*t^0.5
d/dt [(1/9)*t^0.5] = (1/9) * (0.5) * t^(-0.5) * (1)
Now, substituting these derivatives back into the expression, we get:
dV/dt = 2π(t + 9) * (1/9)*t^0.5 + π(t + 9)^2 * (1/9) * (0.5) * t^(-0.5)
Simplifying this expression, we can write:
dV/dt = (2π/9) * (t + 9) * t^0.5 + (π/9) * (t + 9)^2 * 0.5 * t^(-0.5)
Combining like terms, we have:
dV/dt = (2π/9) * (t + 9) * t^0.5 + (π/18) * (t + 9)^2 * t^(-0.5)
Thus, the rate of change of the volume with respect to time is given by:
dV/dt = (2π/9) * (t + 9) * t^0.5 + (π/18) * (t + 9)^2 * t^(-0.5)
The units are in^3/s.
To find the rate of change of the volume with respect to time, we need to use the formula for the volume of a right circular cylinder, which is given by:
V = πr^2h
Where:
V = Volume
r = Radius
h = Height
We are given that the radius of the cylinder is given by t + 9, and the height is (1/9)*(t^0.5).
Let's calculate the volume of the cylinder using these expressions, and then differentiate it with respect to time to find the rate of change.
Step 1: Substitute the given expressions for radius and height into the volume formula.
V = π(t + 9)^2 * (1/9)*(t^0.5)
Step 2: Simplify the expression.
V = π(t + 9)^2 * (1/9)*(t^0.5)
= (π/9)(t + 9)^2 * t^0.5
Step 3: Differentiate the volume formula with respect to time (t).
dV/dt = (π/9) * [(2(t + 9) * t^0.5) + (t + 9)^2 * (0.5 * t^(-0.5))]
Step 4: Simplify the expression.
dV/dt = (π/9) * [2(t + 9) * t^0.5 + 0.5(t + 9)^2 * t^(-0.5)]
= (π/9) * [2(t + 9) * t^0.5 + (t + 9)^2 * 0.5 * t^(-0.5)]
So, the rate of change of the volume with respect to time is given by the expression (π/9) * [2(t + 9) * t^0.5 + (t + 9)^2 * 0.5 * t^(-0.5)] in in^3/s.
volume of cylinder = πr^2 h
= π(1/9)t^.5)(t+9)
= (π/9)t^1.5 + πt
dV/dt = (π/6)t + π