# Modeling with Equations

1. Given the function M(t) = 2t3 - 3t2 - 36t, find the critical values and determine, using both the second derivative test and a sign chart, the nature of these values.

2. A projectile is launched with a velocity of 22 m/s at 50° to the ground. Determine its horizontal and vertical velocities.

3. Two trains start from the same point at the same time, one going east at a rate of 40 km/h and the other going south at 60 km/h, as shown in the diagram at right. Find the rate at which they are separating after 1 h of travel.

4. A professional basketball team plays in a stadium that holds 23,000 spectators. With ticket prices at $60, the average attendance had been 18,000. When ticket prices were lowered to $55, the average attendance rose to 20,000. Based on this pattern, how should ticket prices be set to maximize ticket revenue?

5. Corey is asked to find the maximum value of a function. Not having a complete understanding of the process, Corey decides to find the derivative of the function, set it equal to zero, and solve. The resulting value, Corey reasons, will yield the maximum point. Explain fully why Corey's method is flawed.

6. A 5,000 m_ rectangular area of a field is to be enclosed by a fence, with a moveable inner fence built across the narrow part of the field, as shown.The perimeter fence costs $10/m and the inner fence costs $4/m. Determine the dimensions of the field to minimize the cost.

7. The following table displays the number of HIV diagnoses per year in a particular country.

Year 1997 1998 1999 2000 2001 2002 2003 2004 2005

Diagnoses 2512 2343 2230 2113 2178 2495 2496 2538 2518

a. Using Curve Expert or another curve modelling program, determine an equation that can be used to model this data.

b. Using this model, estimate the number of diagnoses in 1996 and in 2006.

c. At what rate would the number of diagnoses be changing in 2006?

d. Halfway through 2006, the number of new HIV diagnoses was found to be 1232. Assuming this rate stays fairly constant for the remainder of the year, does this new information change the modelling equation? If so, how would this change your answer to part (c)? If you were an advocate for furthering HIV and AIDS research and treatment programs, would you be encouraged or discouraged by these results?

#### Solution Summary

This looks at a variety of problems and illustrates how to use equations to model the problems. It then uses the models to draw conclusions about the data.