Introduction
This WebQuest allows the you to use skills learned in Calculus to solve three problems while embarking on a trip into the world of fantasy. While there, you will delve into the world of pirate ships and treasure hunters. Check out the Task set before you to seek your fame and fortune!
Remember that while you are salvaging treasure in this quest, this is still Calculus. The following is a glossary of terms in Calculus that you will need to know and understand in order to be able to complete the tasks set before you:
- critical value--a point or points on a closed interval where the derivative is 0
- derivative--the slope of the tangent line to a curve
- differentiation--the process of finding a derivative
- extrema--the generic term for maximum or minimum values
- maximum--the largest value on a closed interval
- minimum--the smallest value on a closed interval
- optimization--the process of finding the extrema on a closed interval
Task
You have been trying for ten years to fulfill your life’s dream of being a treasure hunter. You invested your savings in a salvage vessel, hired a crew and set out to seek your fortune. Over the past few centuries, many Spanish galleons were sunk off the Florida coast while transporting treasures from the New World to Spain.
You have enjoyed a modest amount of success over the years, but the big one has always eluded you. After almost ten years of searching for that one ship that contains enough buried gold to allow you to retire in luxury, your dreams have finally come true. You found it. The treasure has been brought to the surface, and is onboard the boat. In order to get it to shore, however, you must overcome three obstacles.
Process
You should be sure that you have consulted the grading rubric and you are clear on all expectations!
Guidelines for Optimization Problems:
-Recognize the type of problem and the types of formulas you will need to use.
-Read the entire problem before attempting to solve.
-Construct a diagram to visually represent the problem.
-Write an appropriate equation or equations to model the problem.
-Get the equation that you want to be minimized or maximized in terms of one variable.
-Differentiate the equation.
-Test the critical values to find the maximum or minimum.
The following are some links that may be helpful in reviewing Optimization properties and examples:
Patrick JMT: Optimization Problem
Mathispower4u: Optimization Problem
Also, you may refer back to your notes on Optimization for extra help/examples if needed!
Task 1:
Despite the excitement of the treasure being located and brought on board the ship, you realize your ship is out of gas. The treasure must be brought to shore in a rowboat. On board the boat is a large piece of metal measuring 16 feet by 12 feet. If a square is cut out of each corner, and the resulting tabs are folded up as sides, a box with an open top will be constructed.
Determine the side of the square to be cut that would result in the largest volume (since you want to carry as much treasure as possible at once!).
Task 2:
Your boat is 4 miles from shore. The treasure is headed for a museum located 10 miles from a point straight from the ship to the shore.
The person rowing the boat can row at a speed of 2 mph. Once on shore, the treasure will need to be carried to the museum. It cannot be transported by motor vehicle, because there are no roads. The people carrying the treasure can walk at a speed of 3 mph. At what point on the shore should the crew row to so that the treasure can reach the museum in the least amount of time?
Task 3:
Congratulations! The treasure is now safely at the museum! The current attendance at the museum averages 6,400 people a week, but once the treasure is put on display, attendance will increase.
The current price for tickets at the museum is $20.00. The head of the museum estimates that for each $2.00 reduction in admission, the attendance will increase by 300 people a week. What price should the museum charge to maximize their weekly revenue? Also, what will that revenue be?
When all three tasks are completed, hand in your work to your teacher!
Evaluation
Your grade will depend on the successful completion of the three tasks your group completes and turns in. The grade is broken down as follows:
Task 1: 20 points
Task 2: 20 points
Task 3: 20 points
Total: 60 points
The following rubric will be used for judging your success with each of the three tasks:
| Item |
Poor 0 - 1 Points |
Adequate 2 - 3 Points |
Outstanding 4 - 5 Points |
Score Earned |
| Figure | No figure is present | A figure is present, but incorrectly drawn | A figure is present and correctly drawn | |
| Formula | Formula(s) missing | Formula(s) present but incorrect | Formula(s) present and correct | |
| Derivative | Function was not differentiated | Function was differentiated but incorrectly | Function was differentiated correctly | |
| Work & Answer | Little to work shown | Some work shown | All work shown |