1 0 3 7 10 12 (days) r(1) (centimeters per day) -6.1 -5.0-4.4 -3.8 -3.5 4. An ice sculpture melts in such a way that it can be modeled as a cone that maintains a conical shape as it decreases in size. The radius of the base of the cone is given by a twice-differentiable function r, where r(t) is measured in centimeters and is measured in days. The table above gives selected values of r'(1), the rate of change of the radius, over the time interval 0 ≤ t ≤ 12. (a) Approximater"(8.5) using the average rate of change of r' over the interval 7 ≤1 ≤ 10. Show the computations that lead to your answer, and indicate units of measure. (b) Is there a time 1. 0 ≤ t ≤ 3, for which r'(t) = -6 ? Justify your answer. (c) Use a right Riemann sum with the four subintervals indicated in the table to approximate the value of 12 r'(1) dr. JO (d) The height of the cone decreases at a rate of 2 centimeters per day. At time 1 = 3 days, the radius is 100 centimeters and the height is 50 centimeters. Find the rate of change of the volume of the cone with respect to time, in cubic centimeters per day, at time t = 3 days. (The volume V of a cone with radius r 1 and height h is V=²h.)
1 0 3 7 10 12 (days) r(1) (centimeters per day) -6.1 -5.0-4.4 -3.8 -3.5 4. An ice sculpture melts in such a way that it can be modeled as a cone that maintains a conical shape as it decreases in size. The radius of the base of the cone is given by a twice-differentiable function r, where r(t) is measured in centimeters and is measured in days. The table above gives selected values of r'(1), the rate of change of the radius, over the time interval 0 ≤ t ≤ 12. (a) Approximater"(8.5) using the average rate of change of r' over the interval 7 ≤1 ≤ 10. Show the computations that lead to your answer, and indicate units of measure. (b) Is there a time 1. 0 ≤ t ≤ 3, for which r'(t) = -6 ? Justify your answer. (c) Use a right Riemann sum with the four subintervals indicated in the table to approximate the value of 12 r'(1) dr. JO (d) The height of the cone decreases at a rate of 2 centimeters per day. At time 1 = 3 days, the radius is 100 centimeters and the height is 50 centimeters. Find the rate of change of the volume of the cone with respect to time, in cubic centimeters per day, at time t = 3 days. (The volume V of a cone with radius r 1 and height h is V=²h.)
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:1
0
3
7
10
12
(days)
r(t)
(centimeters per day)
-6.1 -5.0 -4.4 -3.8 -3.5
4. An ice sculpture melts in such a way that it can be modeled as a cone that maintains a conical shape as it
decreases in size. The radius of the base of the cone is given by a twice-differentiable function r, where r(t) is
measured in centimeters and is measured in days. The table above gives selected values of r'(1), the rate of
change of the radius, over the time interval 0 ≤ t ≤ 12.
(a) Approximater"(8.5) using the average rate of change of r' over the interval 7 ≤ t ≤ 10. Show the
computations that lead to your answer, and indicate units of measure.
(b) Is there a time 1, 0 ≤ ≤ 3, for which r'(t) = -6? Justify your answer.
(c) Use a right Riemann sum with the four subintervals indicated in the table to approximate the value of
[1² rº(1) dt.
(d) The height of the cone decreases at a rate of 2 centimeters per day. At time 1 = 3 days, the radius is
100 centimeters and the height is 50 centimeters. Find the rate of change of the volume of the cone with
respect to time, in cubic centimeters per day, at time t = 3 days. (The volume V of a cone with radius r
1
and height h is V = — zr²h.)
3
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