3. [5pt] Consider a cylindrical tree trunk of radius R. Living cells occupy a thin shell...
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3. [5pt] Consider a cylindrical tree trunk of radius R. Living cells occupy a thin shell (thickness s) just inside the tree bark. The interior of the trunk consists of dead cells that have turned into wood. (a) Let F be the fraction of the trunk vol- ume that is living tissue. Find a formula for F that depends only on R and s. (b) Notice that F and R both change with time as the tree grows, and assume that s does not change. Assuming that the radius of the trunk grows at a constant rate of 2 millimetres per year, we would like to know how F changes as the tree grows. dF Compute the rate of change of F, dt at the instant when the radius R is 10 times the thickness s. 2 (c) According to your calculation, is Fin- creasing or decreasing at that moment? dF = 0. dt What could you tell about the tree's ra- dius at that moment? What would the tree look like? (d) Suppose at some moment, R Top view R Side view 3. [5pt] Consider a cylindrical tree trunk of radius R. Living cells occupy a thin shell (thickness s) just inside the tree bark. The interior of the trunk consists of dead cells that have turned into wood. (a) Let F be the fraction of the trunk vol- ume that is living tissue. Find a formula for F that depends only on R and s. (b) Notice that F and R both change with time as the tree grows, and assume that s does not change. Assuming that the radius of the trunk grows at a constant rate of 2 millimetres per year, we would like to know how F changes as the tree grows. dF Compute the rate of change of F, dt at the instant when the radius R is 10 times the thickness s. 2 (c) According to your calculation, is Fin- creasing or decreasing at that moment? dF = 0. dt What could you tell about the tree's ra- dius at that moment? What would the tree look like? (d) Suppose at some moment, R Top view R Side view
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Related Book For
Vector Mechanics for Engineers Statics and Dynamics
ISBN: 978-0073212227
8th Edition
Authors: Ferdinand Beer, E. Russell Johnston, Jr., Elliot Eisenberg, William Clausen, David Mazurek, Phillip Cornwell
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