Ch. 3 - Derivatives

Briggs - Calculus: Early Transcendentals 3rd Edition

Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook

Briggs 3rd Edition

Ch. 3 - Derivatives

Problem 3.6.25c

Chapter 3, Problem 3.6.25c

Suppose a stone is thrown vertically upward from the edge of a cliff on Earth with an initial velocity of 64 ft/s from a height of 32 ft above the ground. The height (in feet) of the stone above the ground t seconds after it is thrown is s(t) = -16t²+64t+32.
c. What is the height of the stone at the highest point?

Verified step by step guidance

To find the height of the stone at the highest point, we need to determine when the stone reaches its maximum height. This occurs at the vertex of the parabolic function s(t) = -16t² + 64t + 32.

The vertex of a parabola given by the equation s(t) = at² + bt + c can be found using the formula t = -b/(2a). Here, a = -16 and b = 64.

Substitute the values of a and b into the formula: t = -64/(2 * -16). Simplify this expression to find the time t at which the stone reaches its maximum height.

Once you have the value of t, substitute it back into the original height function s(t) = -16t² + 64t + 32 to find the height of the stone at this time.

Calculate s(t) using the value of t found in the previous step to determine the maximum height of the stone above the ground.

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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 of the stone is modeled by a quadratic function, s(t) = -16t² + 64t + 32. Quadratic functions are polynomial functions of degree two, characterized by their parabolic shape. The coefficients determine the direction of the parabola and its vertex, which represents the maximum or minimum point of the function.

Vertex of a Parabola

The highest point of a parabola represented by a quadratic function occurs at its vertex. For a function in the form s(t) = at² + bt + c, the t-coordinate of the vertex can be found using the formula t = -b/(2a). This point gives the time at which the stone reaches its maximum height, which is essential for solving the problem.

Maximizing Height

To find the maximum height of the stone, we substitute the t-coordinate of the vertex back into the height function s(t). This process allows us to determine the maximum value of the function, which corresponds to the highest point the stone reaches during its flight. Understanding this step is crucial for answering the question accurately.

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Textbook Question

Throwing a stone Suppose a stone is thrown vertically upward from the edge of a cliff on Earth with an initial velocity of 32 ft/s from a height of 48 ft above the ground. The height (in feet) of the stone above the ground t seconds after it is thrown is s(t) = -16t²+32t+48.

c. What is the height of the stone at the highest point?

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A rectangular swimming pool 10 ft wide by 20 ft long and of uniform depth is being filled with water.

c. At what rate is the water level rising if the pool is filled at a rate of 10ft³/min?

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62–65. {Use of Tech} Graphing f and f'

c. Verify that the zeros of f' correspond to points at which f has a horizontal tangent line.

f(x) = (x−1) sin^−1 x on [−1,1]

Textbook Question

Vibrations of a spring Suppose an object of mass m is attached to the end of a spring hanging from the ceiling. The mass is at its equilibrium position $y equals 0$ when the mass hangs at rest. Suppose you push the mass to a position $y sub 0$ units above its equilibrium position and release it. As the mass oscillates up and down (neglecting any friction in the system), the position y of the mass after t seconds is $y = y_{0} \cos (t \sqrt{\frac{k}{m}})$ , where $k is greater than 0$ is a constant measuring the stiffness of the spring (the larger the value of $k$ , the stiffer the spring) and $y$ is positive in the upward direction.

Use equation (4) to answer the following questions.

c. How would the velocity be affected if the experiment were repeated with a spring having four times the stiffness ( $k$ is increased by a factor of $4$ )?

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Witch of Agnesi Let y(x²+4)=8 (see figure). <IMAGE>

c. Solve the equation y(x²+4)=8 for y to find an explicit expression for y and then calculate dy/dx.