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ENGINEERING
Design a clepsydra (Egyptian water clock), which is a vessel from which water drains by gravity through a hole in the bottom and that indicates time by the level of the remaining water. Specify the dimensions of the vessel and the size of the drain hole; indicate the amount of water needed to fill the vessel and the interval at which it must be filled. Plot the vessel radius as a function of elevation.
QUESTION
A clepsydra, or water clock, is a glass container with a small hole in the bottom through which water can flow. The “clock” is calibrated for measuring time by placing markings on the container corresponding to water levels at equally spaced times. Let x=f(y) be continuous on the interval [0, b] and assume that the container is formed by rotating the graph of f about the y-axis. Let V denote the volume of water and h the hieght of the water level at time t. Determine V as a function of h.
QUESTION
A clepsydra, or water clock, is a glass container with a small hole in the bottom through which water can flow. The “clock” is calibrated for measuring time by placing markings on the container corresponding to water levels at equally spaced times. Let x=f(y) be continuous on the interval [0, b] and assume that the container is formed by rotating the graph of f about the y-axis. Let V denote the volume of water and h the hieght of the water level at time t. Show that $\frac{d V}{d t}=\pi[f(h)]^{2} \frac{d h}{d t}$
ENGINEERING
The clepsydra, or water clock, was a device used by the ancient civilizations to measure the passage of time by observing the change in the height of water that was permitted to flow out of a small hole in the bottom of a container or tank. Use the differential equation $$ \frac { d h } { d t } = - c \frac { A _ { h } } { A _ { w } } \sqrt { 2 g h } $$ as a model for the height $h$ of water in a tank at time $t$. Assume that $h(0) = 2 \text{ ft}$ corresponds to water filled to the top of the tank, the hole in the bottom is circular with radius $1/32 \text{ in}, g = 32 \text{ ft/s}^2$, and that $c = 0.6$. Suppose that a tank is made of glass and has the shape of a right-circular cylinder of radius $1 \text{ ft}$. Find the height $h(t)$ of the water.