Textbook Question
Find a trigonometric function represented by the graph in the figure. <IMAGE>
Ch. 1 - Functions
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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Table of contents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
Chapter 1, Problem 65e
{Use of Tech} Height and time The height in feet of a baseball hit straight up from the ground with an initial velocity of 64 ft/s is given by h= ƒ(t) = 64t - 16t² where t is measured in seconds after the hit.
e. At what time is the ball at a height of 10 ft on the way down?
Verified step by step guidance1
Step 1: Set up the equation for the height of the ball. We are given that the height function is \( h(t) = 64t - 16t^2 \). We need to find the time \( t \) when the height \( h(t) = 10 \) feet.
Step 2: Substitute 10 for \( h(t) \) in the equation: \( 10 = 64t - 16t^2 \).
Step 3: Rearrange the equation to form a standard quadratic equation: \( 16t^2 - 64t + 10 = 0 \).
Step 4: Use the quadratic formula \( t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) to solve for \( t \), where \( a = 16 \), \( b = -64 \), and \( c = 10 \).
Step 5: Calculate the discriminant \( b^2 - 4ac \) and then find the two possible values for \( t \). Choose the value of \( t \) that corresponds to the ball being on its way down.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Quadratic Functions
The height function h(t) = 64t - 16t² is a quadratic function, which is a polynomial of degree two. Quadratic functions graph as parabolas and can model various physical phenomena, such as projectile motion. Understanding the properties of parabolas, including their vertex and intercepts, is essential for analyzing the height of the baseball over time.
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Solving Quadratic Equations
To find the time when the baseball reaches a height of 10 ft, we need to solve the equation 64t - 16t² = 10. This involves rearranging the equation into standard form and applying methods such as factoring, completing the square, or using the quadratic formula. Mastery of these techniques is crucial for determining specific values of t in quadratic scenarios.
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Projectile Motion
The scenario describes projectile motion, where an object is thrown vertically and affected by gravity. The height of the object over time can be modeled using a quadratic equation, where the initial velocity and gravitational acceleration influence the trajectory. Understanding the principles of projectile motion helps in interpreting the behavior of the baseball as it rises and falls.
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Related Practice
Textbook Question
Prove the following identities.
Textbook Question
Prove the following identities.
Textbook Question
{Use of Tech} Height and time The height in feet of a baseball hit straight up from the ground with an initial velocity of 64 ft/s is given by h= ƒ(t) = 64t - 16t² where t is measured in seconds after the hit.
d. At what time is the ball at a height of 30 ft on the way up?
Textbook Question
{Use of Tech} Height and time The height in feet of a baseball hit straight up from the ground with an initial velocity of 64 ft/s is given by h= ƒ(t) = 64t - 16t² where t is measured in seconds after the hit.
b. Find the inverse function that gives the time t at which the ball is at height h as the ball travels upward. Express your answer in the form t = ƒ⁻¹ (h)
2
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Textbook Question
{Use of Tech} Height and time The height in feet of a baseball hit straight up from the ground with an initial velocity of 64 ft/s is given by h= ƒ(t) = 64t - 16t² where t is measured in seconds after the hit.
c. Find the inverse function that gives the time t at which the ball is at height h as the ball travels downward. Express your answer in the form t = ƒ⁻¹ (h)
2
views