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

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

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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

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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. We need to use: g = 32 ft/s 2 v o = 80 ft/s h o = 200 ft What is the maximum height of the ball? Jeff Bivin -- LZHS

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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? Where is the maximum?Find the vertex…… Vertex is: 300 ft. Jeff Bivin -- LZHS

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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? Where is the maximum?Find the vertex…… Vertex is: 2.5 sec Jeff Bivin -- LZHS

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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? What is the height at the ground?h(t) = 0 Get the decimal approximations: 6.830 sec. Jeff Bivin -- LZHS

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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? 250 feet What height?h(t) = 250 Get the decimal approximations: 0.732 sec & 4.268 sec. Jeff Bivin -- LZHS

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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? What height?h(t) = 400 Wait, what was the maximum height? never Jeff Bivin -- LZHS

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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? What height?h(t) = 50 Get the decimal approximations: 6.453 sec. Jeff Bivin -- LZHS

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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. 20 feet below the ground 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 Jeff Bivin -- LZHS

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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 Get the decimal approximations: 6.972 sec. Jeff Bivin -- LZHS

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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? What time?t = 3 296 ft. Jeff Bivin -- LZHS

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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 !!!! Answer:we don’t know! This formula compares time with height, not horizontal distance. Jeff Bivin -- LZHS

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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

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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

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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? Jeff Bivin -- LZHS Income = Price ● Quantity f(x) = ( 15 + 2x ) ( 200 – 10x ) Define the variable x = number of $2 price increases f(x) = 3000 – 150x + 400x – 20x 2 f(x) = – 20x 2 + 250x + 3000 VERTEX 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 f(x) = income Maximum income = f(x) = $3781.25

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Jeff Bivin -- LZHS

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