# Mathematics

**Q2**.**(64-Ch3)64. Marginal Analysis **Suppose the profit (in cents) from selling **x*** *lb of potatoes is given by **P(X)=15x+25x****2**

1. Find the average rate of change in profit from selling each of the following amounts.

**a. **6 lb to 7 lb **b. **6 lb to 6.5 lb **c. **6 lb to 6.1 lb

Find the marginal profit (that is, the instantaneous rate of change of the profit) from selling the following amounts.

**d. **6 lb **e. **20 lb **f. **30 lb

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**g. **What is the domain of *x*?

**h. **Is it possible for the marginal profit to be negative here? What does this mean?

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**i. **Find the average profit function. (Recall that average profit is given by total profit divided by the number produced, or

**P( x ) = P( x ) / x)**

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**j. **Find the marginal average profit function (that is, the function giving the instantaneous rate of change of the average profit function).

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**k. **Is it possible for the marginal average profit to vary here? What does this mean?

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**l. **Discuss whether this function describes a realistic situation.

**Q3**. (**58-4.1)58. ****Money **The total amount of money in circulation for the years 1950–2009 can be closely approximated by

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**M****(****t)= 0.005209t3 – 2 0.04159t2 – 2 0.3664t + 34.49**

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**1. **where t* *represents the number of years since 1950 and ** M( t )** is in billions of dollars. Find the derivative of

**and use it to find the rate of change of money in circulation in the following years.**

*M( t )***.**

*Source: U.S. Treasury*

*a. 1960 b. 1980 c. 1990 d. 2000*

* *

**e. **What do your answers to parts a–d tell you about the amount of money in circulation in those years?

**Q4.**** (62-4.1)62.Brain Mass **The brain mass of a human fetus during the last trimester can be accurately estimated from the circumference of the head by M ( c ) = c3 /100 – 1500/c,

Where **m( c) ** is the mass of the brain (in grams) and *c *is the circumference (in centimeters) of the head. *Source: Early**Human Development.*

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**a. **Estimate the brain mass of a fetus that has a head circumference of 30 cm.

**b. **Find the rate of change of the brain mass for a fetus that has a head circumference of 30 cm and interpret your results.

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**Q5****.(72-4.1)72.****Velocity **If a rock is dropped from a 144-ft building, its position (in feet above the ground) is given by

S( t ) = -16t2 + 144,

where *t *is the time in seconds since it was dropped.

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**a. **What is its velocity 1 second after being dropped? 2 seconds after being dropped?

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**b. **When will it hit the ground?

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**c. **What is its velocity upon impact?

Q6**.(48-4.2)48. ****Muscle Reaction **When a certain drug is injected into a muscle,the muscle responds by contracting. The amount of contraction, *s *(in millimeters) is related to the concentration of the drug, *x *(in milliliters) by s( x ) = x / m+nx

Where *m *and *n *are constants.

**a. **Find s’( x )

**b. **Find the rate of contraction when the concentration of the drug is 50ml, m =10, and n = 3.

Q7**.(52-4.2) 52. ****Optimal Foraging **Using data collected by zoologist Reto Zach, the work done by a crow to break open a whelk (large marine snail) can be estimated by the function W= (1+ 20 / H – 0.93) H,

Where *H *is the height (in meters) of the whelk when it is dropped. ** Source: Mathematics Teacher**.

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**a. **Find dW | dH

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**b. **One can show that the amount of work is minimized when

dW | dH = 0. Find the value of *H *that minimizes *W*.

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**c. **Interestingly, Zach observed the crows dropping the whelks from an average height of 5.23 m. What does this imply?

Q8**.(64-4.3) ****64. ****Drug Reaction **The strength of a person’s reaction to a certain drug is given by

R (Q) = Q (C – Q/3) 1/2.

where *Q *represents the quantity of the drug given to the patient and *C *is a constant.

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**a. **The derivative R’( Q )is called the *sensitivity *to the drug.

Find R’(Q).

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**b. **Find the sensitivity to the drug if C = 59 and a patient is given 87 units of the drug.

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**c. **Is the patient’s sensitivity to the drug increasing or decreasing where Q = 87?

Q9**.(60-4.4) 60.****Heat Index **The heat index is a measure of how hot it really feels under different combinations of temperature and humidity. The heat index, in degrees Fahrenheit, can be approximated by

**H****(****T)**** =****T – 0.9971e ****0.02086T ****{ 1 – e0.0445****(D -57.2)****},**

where the temperature *T *and dewpoint *D *are both expressed in degrees Fahrenheit. ** Source: American Meteorological Society**.

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**a. **Assume the dewpoint is *D *= 85 F. Find the function *H*(*T*).

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**b. **Using the function you found in part a, find the heat index when the temperature *T *is 80 F.

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**c. **Find the rate of change of the heat index when *T *= 80 F.

Q10**.(68-4.5) ****68. ****Street Crossing **Consider a child waiting at a street corner for a gap in traffic that is large enough so that he can safely cross the street. A mathematical model for traffic shows that if the expected waiting time for the child is to be at most 1 minute, then the maximum traffic flow, in cars per hour, is given by

f (x ) = 29,000(2.322 – log x)/ x ,

Where *x *is the width of the street in feet. Find the maximum traffic flow and the rate of change of the maximum traffic flow with respect to street width for the following values of the street width. ** Source: An Introduction to Mathematical Modeling**.

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**A. ****30 ft ****B. ****40 ft**

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