a. Explanation Given: A basket ball is thrown horizantally at an angle of 45 ° . The center of the ball is ( 0 , 8 ) and hoop is at the point ( 18 , 10 ) and the ball does not passes through the hoop. Definition used: An object travels horizontally on ground and acts only by the gravitational force has an initial position 〈 x 0 , y 0 〉 = 〈 0 , 0 〉 and initial velocity 〈 u 0 , v 0 〉 = 〈 | v 0 | cos α , | v 0 | sin α 〉 . Let the position vector of an object moving in two dimensional space be given by r ( t ) = 〈 x ( t ) , y ( t ) , z ( t ) 〉 for t ≥ 0 . Then, velocity of the object v ( t ) is the derivative of the position vector, r ′ ( t ) = 〈 x ′ ( t ) , y ′ ( t ) , z ′ ( t ) 〉 . The acceleration of the objects is the derivative of the velocity of the object a ( t ) = v ′ ( t ) = r ″ ( t ) . Then, velocity can be obtained from antiderivative of acceleration vector as v ( t ) = ∫ ( a ( t ) ) d t and the position vector can be obtained from antiderivative of velocity vector as r ( t ) = ∫ v ( t ) d t . Calculation: Here the gravitational force acts on the basket ball from horizontal direction ( y -axis ). The initial velocity is 〈 u 0 , v 0 〉 = 〈 | v 0 | cos α , | v 0 | sin α 〉 . 〈 u 0 , v 0 〉 = 〈 | v 0 | cos 45 ° , | v 0 | sin 45 ° 〉 = 〈 | v 0 | 1 2 , | v 0 | 1 2 〉 = 〈 | v 0 | 2 2 , | v 0 | 2 2 〉 Thus, the initail velocity is 〈 | v 0 | 2 2 , | v 0 | 2 2 〉 . The position is an antiderivative of the velocity vector is 〈 | v 0 | 2 2 , − g t + | v 0 | 2 2 〉 . That is, 〈 | v 0 | 2 2 , − 32 t + | v 0 | 2 2 〉 b. To determine The initial velocity of the ball when it released. c. To determine The position function r ( t ) . d. To determine The distance between the center of the basket ball and the front of its hoop. e. To determine The closest distance between the center of the basket ball and the front of its hoop. f. To determine Whether the assumption of the baseketball does not hits front of the hoop is valid or not.

MyLab Math with Pearson eText -- Standalone Access Card -- for Calculus: Early Transcendentals (3rd Edition)

3rd Edition

ISBN: 9780134856926

Author: William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz

Publisher: PEARSON

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Chapter 14, Problem 42RE

To determine

The initial speed of the basket ball.

To determine

The initial velocity of the ball when it released.

To determine

The position function r(t).

To determine

The distance between the center of the basket ball and the front of its hoop.

To determine

The closest distance between the center of the basket ball and the front of its hoop.

To determine

Whether the assumption of the baseketball does not hits front of the hoop is valid or not.

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Chapter 14, Problem 41RE

Chapter 14, Problem 43RE

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Chapter 14 Solutions

MyLab Math with Pearson eText -- Standalone Access Card -- for Calculus: Early Transcendentals (3rd Edition)

Ch. 14.1 - Restrict the domain o f the vector function in...Ch. 14.1 - Explain why the curve in Example 5 lies on the...Ch. 14.1 - How many independent variables does the function...Ch. 14.1 - How many dependent scalar variables does the...Ch. 14.1 - Prob. 3E Ch. 14.1 - Prob. 4E Ch. 14.1 - How do you evaluate limtar(t), where r(t) = f(t),...Ch. 14.1 - How do you determine whether r(t) = f(t) i + g(t)...Ch. 14.1 - Find a function r(t) for the line passing through...Ch. 14.1 - Find a function r(t) whose graph is a circle of...

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