A tank has the shape of the parabola y = ax2 (where a is a constant) revolved around the y-axis. Water drains from a hole of area B m² at the bottom of the tank. (a) Show that the water level at time t is За В 28 2/3 3/2 y(t) = (Yo 2л where yo is the water level at time t = 0. (b) Show that if the total volume of water in the tank has volume V at time t = 0, then yo = /2aV /a. Hint: Compute the volume of the tank as a volume of rotation. (c) Show that the tank is empty at time 27 V3 1/4 le = 3B /8 We see that for fixed initial water volume V, the time te is proportional to a-1/4. A large value of a corresponds to a tall thin tank. Such a tank drains more quickly than a short wide tank of the same initial volume.
A tank has the shape of the parabola y = ax2 (where a is a constant) revolved around the y-axis. Water drains from a hole of area B m² at the bottom of the tank. (a) Show that the water level at time t is За В 28 2/3 3/2 y(t) = (Yo 2л where yo is the water level at time t = 0. (b) Show that if the total volume of water in the tank has volume V at time t = 0, then yo = /2aV /a. Hint: Compute the volume of the tank as a volume of rotation. (c) Show that the tank is empty at time 27 V3 1/4 le = 3B /8 We see that for fixed initial water volume V, the time te is proportional to a-1/4. A large value of a corresponds to a tall thin tank. Such a tank drains more quickly than a short wide tank of the same initial volume.
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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Transcribed Image Text:A tank has the shape of the parabola y = ax2 (where a is a constant) revolved around the y-axis. Water drains from
a hole of area B m² at the bottom of the tank.
(a) Show that the water level at time t is
За В 28
2/3
3/2
y(t) = (Yo
2л
where yo is the water level at time t = 0.
(b) Show that if the total volume of water in the tank has volume V at time t = 0, then yo = /2aV /a. Hint: Compute
the volume of the tank as a volume of rotation.
(c) Show that the tank is empty at time
27 V3 1/4
le =
3B /8
We see that for fixed initial water volume V, the time te is proportional to a-1/4. A large value of a corresponds to a tall
thin tank. Such a tank drains more quickly than a short wide tank of the same initial volume.
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