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April 4, 2008 Project II of Math 181 Name:
You must do this project by yourselves, and you are not allowed to discuss with others about the problems. The answer for each problem must be neatly and clearly written down. Only solution will cause no points. DUE ON MAY 14 IN CLASS. (Total is 60 points)
1. (10 points) Suppose the particle potential changes with respect to time such that
Φ(t) = 3 t^6 + 2t^2 − 5 t + 4 t^2
(a) What is the instantaneous rate of change of the potential of f when t = 1? (b) Find the tangent line to the potential function at t = 2? (c) Physically, the potential is the integral of the external and internal forces. So, the derivative of the potential is the rate of change of the total forces? What is the rate of change for the total forces at t = 1?
2. (10 points) Temperature is commonly measured in two different scales–the Fahrenheit scale
with units oF, and the Celsius scale with units oC. Water freezes at 0oC, which in Fahrenheit is recorded as 32oF. Water boils at 100oC, while in Fahrenheit the boiling temperature is 212oF. If F is the temperature in oF and C is the temperature in oC, then a straight line gives the relationship. (a) Find the equation for F in terms of C. (b) The usual body temperature of a human is 98. 6 oF. What is the usual body temperature measure in degrees Celsius? (c) A recipe in a French cookbook tells you to bake at 200oC. Your American oven uses degrees Fahrenheit. At what temperature should you set your oven?
3. (10 points) A yield of corn is a function
Y =
10 F
2 + F 2
for F ≥ 0 where F is the amount of fertilizer in the solid in tones/acre and Y is measure in tones/acre. (a) Find the level of fertilizer that will maximize the yield. (b) Find the maximum yield.
6. (10 points) A population of deer oscillates due to periodic predation. Tje population od
deer t years after the base time is
p(t) = 24 + 11 sin(0. 898 t − 0 .211)
thousand deers. (a) What is population 3.4 years after the base time? (b) What is the minimum time that there is a maximal population of deer? And, what is the minimum time that there is a minimum population of deer? (c) What is the maximum number of deer in the population? (d) Tell the first two successive positive times t when the number of deer is maximized?
