Ch. 3 - Derivatives

Briggs - Calculus: Early Transcendentals 3rd Edition

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Briggs 3rd Edition

Ch. 3 - Derivatives

Problem 3.6.23c

Chapter 3, Problem 3.6.23c

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?

Verified step by step guidance

To find the height of the stone at its highest point, we need to determine when the stone reaches its maximum height. This occurs at the vertex of the parabola represented by the quadratic function s(t) = -16t² + 32t + 48.

The vertex of a parabola given by the equation ax² + bx + c can be found using the formula t = -b/(2a). In this case, a = -16 and b = 32.

Substitute the values of a and b into the vertex formula: t = -32/(2 * -16). This will give you 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 + 32t + 48 to find the height of the stone at this time.

Evaluate s(t) using the calculated value of t 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, which is a polynomial of degree two. Quadratic functions have a parabolic shape and can be expressed in the form s(t) = at² + bt + c, where a, b, and c are constants. The vertex of the parabola represents the maximum or minimum point, which is crucial for determining the highest point of the stone's trajectory.

Vertex of a Parabola

The vertex of a parabola given by the function s(t) = -16t² + 32t + 48 can be found using the formula t = -b/(2a). In this case, 'a' is -16 and 'b' is 32. The vertex provides the time at which the stone reaches its maximum height, and substituting this time back into the height function gives the maximum height.

Maximizing Functions

To find the maximum height of the stone, we need to evaluate the function at the vertex. This involves calculating the height at the time derived from the vertex formula. Understanding how to maximize a function is essential in calculus, as it applies to various real-world scenarios, including projectile motion like that of the thrown stone.

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