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# Math and Science Solutions for Businesses/Calculus

Question
given l(v)=8.1-.067v
l(v)= miles per gallon, v = velocity
a. if fuel costs 3.80 per gallon, use the formula for l(v) to determine a function, F(v) which gives fuel cost as a function of driving speed, v.
b. Assume you pay your drivers an hourly wage of 22 per hour, use this fact to find a function W(v) which gives wage cost as a function of driving speed.
c. the total cost, C = m(v), is the sum of the functions m(v) = F(v) + W(v). Determine the driving speed, v, between 45 and 65, that minimizes the total cost, use the second derivative test to classify any critical points.
Thank you so much i have had trouble with this for days..

The trick to this question seems to be to realize that the "cost" really refers to a cost per mile or per time. So for part a, assuming we want \$/mile, we have

p = \$3.80/gal and I(v) = miles/gal

which need to be combined to give cost/mi = F(v) = \$/mi = p/I(v) = (\$/gal)/(mi/gal).

Similarly, for the drivers earning q = \$22/hour, we get

cost/mi = W(v) = q/v = (\$/hr)/(mi/hr) = \$/mi.

It is important to note that in order to get the correct dimension for the cost function, i.e., \$/mi, I took the 2 quantities I had to work with, namely for part a the price per gallon, \$/gal, and miles/gal, and combined them in such a way as to get the proper dimensions. Checking dimensions in a problem like this is a really good habit to get into.

Back to the problem, the sum is thus

m(v) = F(v) + W(v) = 3.8/(8.1 - (0.067)v + 22/v.

You then need to take the derivative of m(v), which is easy, and set it equal to 0. You will get a quadratic equation in v which you can solve using the quadratic equation. Doing this I get a value of v = 46.3 for m(v) a minimum.

Taking the second derivative of m(v) and evaluating it at this value of v you should get a positive result, which means the slope is increasing with v, corresponding to a minimum.
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Math and Science Solutions for Businesses

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#### Randy Patton

##### Expertise

Questions regarding application of mathematical techniques and knowledge of physics and engineering principles to product and services design, optimization, prediction, feasibility and implementation. Examples include sales and product performance projections based on math/physics models in addition to standard regression; practical and cost effective sensor design and component configuration; optimal resource allocation using common tools (eg., MS Office); advanced data analysis techniques and implementation; simulation and "what if" analysis; and innovative applications of remote sensing.

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26 years as professional physical scientist and project manager for elite research company providing academic quality basic and applied research for government and defense industry clients (currently retired). Projects I have been involved in include: - Notional sensor performance predictions for detecting underwater phenomena - Designing and testing guidance algorithms for multi-component system - Statistical analysis of ship tracking data and development of anomaly detector - Deployed vibration sensors in Arctic ice floes; analysis of data - Developed and tested ocean optical instrument to measure particles - Field testing of protoype sonar system - Analysis of synthetic aperture radar system data for ocean surface measurements - Redesigned dust shelters for greeters at Burning Man Festival Project management with responsibility for allocation and monitoriing of staff and equipment resources.

Publications
“A Numerical Model for Low-Frequency Equatorial Dynamics” (with Mark A. Cane), J. of Phys. Oceanogr., 14, No. 12, pp. 18531863, December 1984.

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MIT, MS Physical Oceanography, 1981 UC Berkeley, BS Applied Math, 1976

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