- Solve direct variation problems.
- Solve inverse variation problems.
- Solve problems involving joint variation.
Animal Kingdom Adventures
Welcome to the world of animal biology! From the fastest cheetahs to the largest elephants, the animal kingdom is full of fascinating mathematical relationships. Zoologists, wildlife photographers, and conservationists use variation models to understand animal behavior, predict population changes, and design better wildlife habitats.
The Great Migration – Cheetah Speed Analysis
Suppose that wildlife researchers have discovered that a cheetah’s hunting success rate varies directly with the square of its sprint speed. When a cheetah runs at 60 mph, it has an 85% success rate catching gazelles.
Find the speed-success relationship and predict hunting outcomes at different speeds:
- 45mph
- 70mph
- 30mph
Elephant Water Needs
Suppose that a different set of researchers found that the daily water consumption of elephants varies inversely with the square of the distance they must travel to reach water sources. When the watering hole is 4 kilometers away, elephants drink 150 gallons per day.
Create a water consumption model for different distances to water sources.
- At 2km
- At 6km
- At 8km
Penguin Colony Dynamics
Now, suppose that marine biologists studying emperor penguins in Antarctica discovered that fish catch rates vary in a complex way. The number of fish caught per hour [latex]F[/latex] varies directly with the square of the number of penguins in a hunting group [latex]P[/latex] and inversely with the depth [latex]D[/latex] they must dive (in meters).
When 6 penguins hunt together at a depth of 18 meters, they catch 24 fish per hour. Find the model and predict different scenarios.
- 4 penguins at 12m depth
- 8 penguins at 24m depth
- 3 penguins at 9m depth