Answer:
0.325 ft/min
Step-by-step explanation:
To solve we must follow the following steps:
First, the volume of a cone is given by the equation:
Vc = (1/3) * Pi * (r ^ 2) * h
where r is the radius and h is the height.
they tell us that the height is equal to twice the radius, therefore
h = 2r; r = h / 2
replacing r in the volume of the cone:
Vc = (1/3) * Pi * ((h / 2) ^ 2) * h
solving we have:
Vc = (1/3) * Pi * (1/4) * (h ^ 2) * h
Vc = (1/12) * Pi * h ^ 3
They give us the change in volume with respect to time, that is, dV / dt = 50
It can also be expressed in the following way:
dV / dt = (dV / dh) * (dh / dt)
We know V, so we can derive with respect to h, we are like this:
dV / dh = 3 * (1/12) * Pi * h ^ 2
dV / dh = (1/4) * Pi * h ^ 2
we know that h is equal to 14, therefore:
dV / dh = (1/4) * Pi * 14 ^ 2
dV / dh = (1/4) * 3.14 * 196 = 153.86
Now if we replace
dV / dt = (dV / dh) * (dh / dt)
50 = 153.86 * (dh / dt)
(dh / dt) = 50/158.36
(dh / dt) = 0.3249
This means that the change in height with respect to time is 0.325 feet per minutes.
Answer:
The height of the pile is increasing at a rate of 0.325 ft/min
Step-by-step explanation:
Given
1) (dV/dt) = 50 ft³/min
2) The shape formed is a cone with diameter = height.
That is, d = h
2r = h
r = (h/2)
Find (dh/dt) when h = 14 ft
Volume of a cone = (πr²h/3)
with r = (h/2)
Volume of the cone = [π(h/2)²h/3]
V = (πh³/12)
But
(dh/dt) = (dh/dV) × (dV/dt)
V = (πh³/12)
(dV/dh) = (πh²/4)
(dh/dV) = (4/πh²)
(dh/dt) = (dh/dV) × (dV/dt)
(dh/dt) = (4/πh²) × 50 = (200/πh²)
when h = 14 ft
(dh/dt) = (200/π×14²)
(dh/dt) = 0.325 ft/min
Hope this Helps!!!
