Answer to Question #282208 in Calculus for jem

Question

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1. water flows into a conical vessel 18cm deep and 10cm in a diameter. if the rate at which the water surface is rising is 27.52mm/s, how fast is the water flowing into the conical vesel in m^3/s when the initial depth of water is 12cm?

Answer 1

1.

Let $V$ denote the volume of the water in the tank.

Let $h$ denote the height of the water in the tank.

Let $r$ denote the radius of the water in the tank.

We are given that $V=\dfrac{1}{3}\pi r^2 h$

Since corresponding parts of similar triangles are proportional, then

$$\dfrac{r}{h}=\dfrac{10/2}{18}=>r=\dfrac{5}{18}h$$

$$V=\dfrac{1}{3}\pi (\dfrac{5}{18}h)^2 h$$

$$V=\dfrac{25}{972}\pi h^3$$

Differentiate both sides with respect to $t$

$$\dfrac{dV}{dt}=\dfrac{25}{324}\pi h^2\dfrac{dh}{dt}$$

Given $\dfrac{dh}{dt}=0.02752\ m/s, h=0.12\ m$

$$\dfrac{dV}{dt}=\dfrac{25}{324}\pi (0.12)^2(0.02752)\approx9.6063\times10^{-5}\ m^3/s$$