Quadratic Applications ------------------------------- Vertical Motion & Profit / Income

1. Quadratic Applications-------------------------------Vertical Motion&Profit / Income By: Jeffrey Bivin Lake Zurich High School jeff.bivin@lz95.org Last Updated: November 30, 2007

2. Vertical Motion • Compares the height of an object with the time in flight. g = force of gravity: 32ft/sec or 9.8 m/sec vo = initial velocity ho = initial height Jeff Bivin -- LZHS

3. A ball is thrown into the air from the top of a 200 foot tall building with an initial upward velocity of 80 ft/sec. • What is the maximum height of the ball? • When will the ball reach the maximum height? • When will the ball return to the ground? • When will the ball be at a height of 250 feet? • When will the ball be at a height of 400 feet? • When will the ball be at a height of 50 feet? • If the ball lands in a 20 foot deep pit, when will the ball hit the bottom of the pit? • What will be the height of the ball in 3 seconds? • How far from the building will the ball land? Jeff Bivin -- LZHS

4. A ball is thrown into the air from the top of a 200 foot tall building with an initial upward velocity of 80 ft/sec. • What is the maximum height of the ball? We need to use: g = 32 ft/s2 vo = 80 ft/s ho = 200 ft ? 200 ft Jeff Bivin -- LZHS

5. A ball is thrown into the air from the top of a 200 foot tall building with an initial upward velocity of 80 ft/sec. • What is the maximum height of the ball? 300 ft. Where is the maximum? Find the vertex…… Vertex is: Jeff Bivin -- LZHS

6. A ball is thrown into the air from the top of a 200 foot tall building with an initial upward velocity of 80 ft/sec. • When will the ball reach the maximum height? 2.5 sec Where is the maximum? Find the vertex…… Vertex is: Jeff Bivin -- LZHS

7. A ball is thrown into the air from the top of a 200 foot tall building with an initial upward velocity of 80 ft/sec. • When will the ball return to the ground? 6.830 sec. What is the height at the ground? h(t) = 0 200 ft Get the decimal approximations: Jeff Bivin -- LZHS

8. A ball is thrown into the air from the top of a 200 foot tall building with an initial upward velocity of 80 ft/sec. • When will the ball be at a height of 250 feet? 0.732 sec & 4.268 sec. 250 feet What height? h(t) = 250 ? 200 ft Get the decimal approximations: Jeff Bivin -- LZHS

9. A ball is thrown into the air from the top of a 200 foot tall building with an initial upward velocity of 80 ft/sec. • When will the ball be at a height of 400 feet? never What height? h(t) = 400 so? Wait, what was the maximum height? Jeff Bivin -- LZHS

10. A ball is thrown into the air from the top of a 200 foot tall building with an initial upward velocity of 80 ft/sec. • When will the ball be at a height of 50 feet? 6.453 sec. What height? h(t) = 50 Get the decimal approximations: Jeff Bivin -- LZHS

11. A ball is thrown into the air from the top of a 200 foot tall building with an initial upward velocity of 80 ft/sec. • If the ball lands in a 20 foot deep pit, when will the ball hit the bottom of the pit? What height? h(t) = -20 200 ft 20 feet below the ground Jeff Bivin -- LZHS

12. A ball is thrown into the air from the top of a 200 foot tall building with an initial upward velocity of 80 ft/sec. • If the ball lands in a 20 foot deep pit, when will the ball hit the bottom of the pit? 6.972 sec. What height? h(t) = -20 Get the decimal approximations: Jeff Bivin -- LZHS

13. A ball is thrown into the air from the top of a 200 foot tall building with an initial upward velocity of 80 ft/sec. • What will be the height of the ball in 3 seconds? 296 ft. What time? t = 3 Jeff Bivin -- LZHS

14. A ball is thrown into the air from the top of a 200 foot tall building with an initial upward velocity of 80 ft/sec. • How far from the building will the ball land? Wait !!!! This formula compares time with height, not horizontal distance. Answer: we don’t know! 200 ft ? Jeff Bivin -- LZHS

15. That was FUN! Let's do one more! Jeff Bivin -- LZHS

16. A diver dives off a 3 meter diving board into a pool with an initial upward velocity of 3.5 m/sec. • What is the maximum height of the diver? • When will the diver reach his/her maximum height? • When will the diver splash into the water? • What will be the height of the diver in 1 second? Jeff Bivin -- LZHS

17. A diver dives off a 3 meter diving board into a pool with an initial upward velocity of 3.5 m/sec. • What is the maximum height of the diver? • When will the diver reach his/her maximum height? • When will the diver splash into the water? • What will be the height of the diver in 1 second? 3.625 meters 0.358 sec. 1.217 sec. 1.6 meters Jeff Bivin -- LZHS

18. That was FUN! Let's do one more! Jeff Bivin -- LZHS

19. A taxi service operates between two airports transporting 200 passengers a day. The charge is $15.00. The owner estimates that 10 passengers will be lost for each $2 increase in the fare. What charge would be most profitable for the service? What is the maximum income? VERTEX Income = Price ● Quantity Define the variable x = number of $2 price increases f(x) = ( 15 + 2x ) ( 200 – 10x ) f(x) = 3000 – 150x + 400x – 20x2 f(x) = income f(x) = – 20x2 + 250x + 3000 f(6.25) = – 20(6.25)2 + 250(6.25) + 3000 f(6.25) = 3781.25 Vertex is: So, price = (15 + 2x) = (15 + 2(6.25)) = 15 + 12.5 = $27.50 Maximum income = f(x) = $3781.25 Jeff Bivin -- LZHS

20. That was FUN! Jeff Bivin -- LZHS