CALCULUS I

Boggess 15.1: The Ant and the Blade of Grass (Lab Project)

An ant is walking (to the right) over its ant mound, whose height (in inches) is given by the function: h(x) = (x^2/16 - 2*x + 80) /(x^2 /16 - 2*x + 20)^2 . Nearby there is a blade of grass, which is located as the line segment from (32, 1/5) to (32, 8). The goal in this lab is to find the point where the ant first sees the blade of grass. You can assume that the ant's line of sight is the tangent line to the ant mound. The following series of questions will lead you to the solution. Answer them on the lab report form.

1.
Define the function h that gives the height of the ant mound, using an arrow definition. Define the line segment occupied by the blade of grass and name it grass . Plot the ant mound and the blade of grass on the same graph.

Part 1

2.
Compute the derivative of h and name it Dh . On the lab report, record Maple input but not the output.

Part 2

3.
Compute the tangent line to y=h(x) at x=12.5 and define it as a function htan using an arrow definition. Plot the ant mound, the blade of grass, and the ant's line of sight when the ant is at x=12.5. Can it see the blade of grass? Find the height H where the tangent line crosses the line x=32 by evaluating htan(32) .

Part 3

4.
Compute the tangent line to y=h(x) at x=15.5 and define it as a function htan using an arrow definition. Plot the ant mound, the blade of grass, and the ant's line of sight when the ant is at x=15.5. Can it see the blade of grass? Find the height H where the tangent line crosses the line x=32 by evaluating htan(32) .

Part 4

5.
We can now see that, when the ant is at some position x=a between 12.5 and 15.5, it can first see the top of the blade of grass. We want to find a. So, compute the tangent line to y=h(x) at x=a for a variable a. Define the tangent line at x=a as a function htan using an arrow definition.

Part 5

6.
You can no longer plot the tangent line because its formula contains a variable, namely a . However, you can still find the height H where the tangent line crosses the line x=32 by evaluating htan(32) . When this height H equals the height of the blade of grass, the ant can just begin to see the blade of grass. Use Maple's fsolve command to solve for the value of a where H equals the height of the blade of grass. (You may need to specify a range for a in the fsolve command.) Denote the solution by A .

Part 6

7.
For the value A found in problem 6, compute the tangent line to y=h(x) at x=A and define it as a function htan using an arrow definition. Plot the ant mound, the blade of grass, and the ant's line of sight when the ant is at x=A. Can it see the blade of grass? Find the height H where the tangent line crosses the line x=32 by evaluating htan(32).

Part 7

8.
10% Extra Credit. There is a second solution to the equation H=8. What is wrong with this solution?

Part 8

After so much hard work we are now ready for a movie.

Ant Movie