The volume, V cm3 , of a cone height h is
pi x h^3 / 12
If h increases at a constant rate
of 0.2 cm/sec and the initial height is 2 cm, express V in terms of t and find the rate of
change of V at time t.
h(t) = 2 + 0.2t
so, plug that in to express v(t)
Then find dv/dt
Or, since
dv/dt = pi/4 h^2 dh/dt,
plug in h(t) and dh/dt=0.2 to express dv/dt
To express V in terms of t, we know that h is increasing at a constant rate of 0.2 cm/sec.
Let's denote t as the time in seconds and V as the volume in cm^3.
Since the initial height is 2 cm, we can express h as:
h = 2 + 0.2t
To express V in terms of t, we substitute the value of h into the volume equation:
V = (π * h^3) / 12
V = (π * (2 + 0.2t)^3) / 12
Now, to find the rate of change of V at time t, we can take the derivative of V with respect to t:
dV/dt = (π/12) * 3 * (2 + 0.2t)^2 * 0.2
dV/dt = (π/12) * 3 * (2 + 0.2t)^2 * 0.2
Simplifying further, we get:
dV/dt = (π/12) * 0.6 * (2 + 0.2t)^2
Therefore, the rate of change of V at time t is given by:
dV/dt = (π/12) * 0.6 * (2 + 0.2t)^2
To express V in terms of t, we need to find an equation that relates the volume of the cone to time.
Let's start by finding the equation that relates the height of the cone to time. We know that the initial height, h0, is 2 cm, and the height, h, increases at a constant rate of 0.2 cm/sec. Thus, we can express the height as h = h0 + rt, where r is the rate of change of height and t is the time.
The expression for the cone's volume is V = (π * h^3) / 12. Substituting h = h0 + rt into this equation, we get:
V = (π * (h0 + rt)^3) / 12
Simplifying this equation further, we have:
V = (π * (h0^3 + 3h0^2 * rt + 3h0 * r^2 * t^2 + r^3 * t^3)) / 12
Now, to find the rate of change of V at time t, we need to differentiate the volume equation with respect to time t. Let's calculate the derivative:
dV/dt = (π * (3h0^2 * r + 6h0 * r^2 * t + 3r^3 * t^2)) / 12
Simplifying this, we have:
dV/dt = (π * r * (3h0^2 + 6h0r * t + 3r^2 * t^2)) / 4
Thus, the expression for the rate of change of V at time t is:
dV/dt = (π * r * (3h0^2 + 6h0r * t + 3r^2 * t^2)) / 4