# Modeling

## ​​A recurring strand in Exeter's math curriculum

Mathematical models arise in a variety of settings throughout our problem sets. Here are a few examples.

Math 1: Chris does a lot of babysitting. When parents drop off their children and Chris can supervise them at home, the hourly rate is \$3. If Chris has to travel to the child’s home, there is a fixed charge of \$5 for transportation in addition to the \$3 hourly rate.

• Graph y = 3x and y = 3x + 5. What do these lines have to do with the babysitting context? What feature do they have in common? How do they differ?
• What does the graph of y = 3x + 6 look like? What change in the babysitting context does this line suggest?

(Math 1, #193) Math 2: Peyton’s workout today is to run repeatedly up a steep grassy slope, represented by ADFC in the diagram. The workout loop is AGCA, in which AG requires exertion and GCA is for recovery. Point G was chosen on the ridge CF to make the slope of the climb equal 1/5. Given that ADEB and BEFC are rectangles, ABC is a right angle, AD = 240, DE = 150, and EF = 50, find the distance from point G to point C.

(Math 2, 75#9)

Math 3: Suppose that three PEA students know a certain rumor at noon, that there are 1000 students in all, and that any student will try to pass a rumor along to one more student each hour. This suggests the model , where is the number of students who know the rumor hours after noon. It is given that = 3. The fraction in the recursive equation represents that portion of the student body that does not know the rumor. Calculate , , and . Your answers will suggest the equation , which describes 100% growth. Explain why this model becomes less realistic as n increases.

(Math 3, 77#3) Math 4: After being thrown from the top of a tall building, a projectile follows the path (x, y) = (60t, 784 − 16t2), where x and y are in feet and t is in seconds. The Sun is directly overhead, so that the projectile casts a moving shadow on the ground beneath it, as shown in the figure.

When t = 1, how fast is the shadow moving? How fast is the projectile losing altitude? How fast is the projectile moving?

What is the altitude of the projectile when t = 2? What is the altitude of the projectile a little later, when t = 2 + k? How much altitude is lost during this k-second interval? At what rate is the projectile losing altitude during this interval?

(Math 4, 11#6)

Other strands that arise throughout our problem sets: