Explanation Given: At t = 0 , a conical tank of height 300 cm and top radius 100 cm is filled with water. Water leaks trough a hole in the bottom of area B = 3 cm 2 . Let y ( t ) be the water level at time t. Calculation: a. By using the property of similar triangles, we get: y 300 = r 100 , where r is the radius of the tank at height y . So, r = y 3 Therefore, the tank’s cross-sectional area at height y is A ( y ) : A ( y ) = π r 2 = π ( y 3 ) 2 = π 9 y 2 Thus, it is shown that the tank’s cross-sectional area at height y is A ( y ) = π 9 y 2 . b. It is given that a hole has an area of B = 3 cm 2 . According to Toricelli’s Law, the velocity of the water flowing through a hole is: v = − 2 g y , where g = 9.8 m s 2 . So, the differential equation for the height of the water in the tank is: d y d t = B v ( y ) A ( y ) , where A ( y ) is the cross-sectional area of the tank at height y . Now, d y d t = B v ( y ) A ( y ) = B ( − 2 g y ) A = 3 × 10 − 4 ( − 2 ( 9.8 ) y ) π ( 100 ) 2 × 10 − 4 = − 13.28 × 10 − 4 y π = − 4.23 × 10 − 4 y So, the separation of variables and then integrating: d y y = − 4.23 × 10 − 4 d t ∫ d y y = ∫ − 4.23 × 10 − 4 d t 2 y = − 4.23 × 10 − 4 t + C y = − 4...

4th Edition

ISBN: 9781319323394

Author: Rogawski

Publisher: MAC HIGHER

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Chapter 10.1, Problem 46E

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a. Show that the tank’s cross-sectional area at height y is $A (y) = \frac{π}{9} y^{2}$ .

b. Find and solve the differential equation satisfied by $y (t)$ .

c. How long does it take for the tank to empty?

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Chapter 10.1, Problem 45E

Chapter 10.1, Problem 47E

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Chapter 10 Solutions

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We need to a. Show that the tank’s cross-sectional area at height y is A ( y ) = π 9 y 2 . b. Find and solve the differential equation satisfied by y ( t ) . c. How long does it take for the tank to empty?

Solution Summary: The author explains how to determine a tank's cross-sectional area at height y by using the property of similar triangles.

Concept explainers

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We need to

a. Show that the tank’s cross-sectional area at height y is $A (y) = \frac{π}{9} y^{2}$ .

b. Find and solve the differential equation satisfied by $y (t)$ .

c. How long does it take for the tank to empty?

Want to see the full answer?

Chapter 10 Solutions

We need to a. Show that the tank’s cross-sectional area at height y is A ( y ) = π 9 y 2 . b. Find and solve the differential equation satisfied by y ( t ) . c. How long does it take for the tank to empty?

Solution Summary: The author explains how to determine a tank's cross-sectional area at height y by using the property of similar triangles.

Concept explainers

Videos

We need to a. Show that the tank’s cross-sectional area at height y is A(y)=π9y2. b. Find and solve the differential equation satisfied by y(t). c. How long does it take for the tank to empty?

Want to see the full answer?

Chapter 10 Solutions

We need to

a. Show that the tank’s cross-sectional area at height y is $A (y) = \frac{π}{9} y^{2}$ .

b. Find and solve the differential equation satisfied by $y (t)$ .

c. How long does it take for the tank to empty?