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Math 171 Group Worksheet over Chapter 5Ask your instructor or classmates for help, but not the Math Learning Center.

Problem 1. We want to approximate the volume of the wine bottleshown below. This wine bottle was chosen because it has an interestingshape that cannot be considered as a solid of revolution. You will learntechniques to compute volumes of solids of revolution in M172.

Slicing the bottle parallel to its bot-tom, we see that the cross sectionsare circular. Thus, we can approx-imate the volume using a sectionmethod. The idea is to break theobject up into pieces that we cancompute the volume of.

Each cross section is a circle and the dimensions of the widths of the bottleare given every 2 centimeters for the bottom 18 centimeters of the bottle inFigure below. Use this information to estimate the volume of the bottom18 centimeters of the bottle.

18.0 cm

5.5

cm

7.5

cm

7.5

cm

7.0

cm

5.5

cm

4.0

cm

8.0

cm

2.5

cm

2.0

cm

2.0

cm

Find an over estimation and an under estimation of the true volume.

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Problem 2. The table below gives the expected growth rate, g(t),in ounces per week, of the weight of a baby in its first 54 weeks of life.Assume for this problem that g(t) is a decreasing function.

Week t 0 9 18 27 36 45 54

growth rate g(t) 6 6 4.5 3 3 3 2

a. Using six subdivisions, find an overestimate and underestimate for thetotal weight gained by a baby over its first 54 weeks of life.

b. How frequently over the 54 week period would you need data for g(t)to be measured to find overestimates and underestimates for the totalweight gain over this time period that differ by 0.5 lb (8 oz)? (See thehandout on Estimating Error in Riemann Sums)

Problem 3. Show that F (x) = tan2(x) and G(x) = sec2(x) have thesame derivative. What can you conclude about the relation between F

and G? Verify this conclusion directly.

Problem 4. Recently, a team of Mechanical Engineers (Li et al., PNASvol 110 (50), 20023-20027, 2013) gave an argument that the huge stonesthat make up parts of China’s Forbidden City were transported alongartificial ice paths lubricated with water. Part of the argument involves acalculation of the total amount of heat due to friction that diffuses intothe ice-covered ground. They needed this to see if friction alone wouldmaintain a ice-water interface or if the ancients needed to pour water ontothe road to maintain the slick water-ice interface needed to transport thehuge stones, 123 tons, which where reported to be transported 70 km in28 days.

They assumed that the rate at which heat diffusing into the ice-coveredroad per unit area is given by

dQ

dt=k(T0 − T1)√

παt

where k, T0, T1, α and of course π are all constants, and t is the contacttime.

If Q(0) = 0 and τ is the total contact time, compute how much heat perunit area diffused into the road, i.e., find Q(τ) assuming Q(0) = 0.