{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)

검증된 단계별 안내

Step 1: Start with the given function for height: \( h = f(t) = 64t - 16t^2 \).

Step 2: To find the inverse function, solve for \( t \) in terms of \( h \). Begin by setting \( h = 64t - 16t^2 \).

Step 3: Rearrange the equation to form a standard quadratic equation: \( 16t^2 - 64t + h = 0 \).

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 \( c = h \).

Step 5: Since we are interested in the time when the ball is traveling upward, choose the solution with the positive square root: \( t = \frac{64 + \sqrt{64^2 - 4 \cdot 16 \cdot h}}{2 \cdot 16} \).

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이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Quadratic Functions

The height function h(t) = 64t - 16t² is a quadratic function, which is characterized by its parabolic shape. Quadratic functions can be expressed in the standard form ax² + bx + c, where a, b, and c are constants. In this case, the coefficient of t² is negative, indicating that the parabola opens downward, which is typical for projectile motion.

Inverse Functions

An inverse function essentially reverses the effect of the original function. For a function f(t), the inverse f⁻¹(h) allows us to find the input t for a given output h. To find the inverse of a quadratic function, we typically solve for t in terms of h, which may involve rearranging the equation and applying the quadratic formula.

Projectile Motion

Projectile motion describes the motion of an object thrown into the air, influenced only by gravity after its initial launch. The height of the object over time can be modeled by a quadratic equation, where the initial velocity and gravitational acceleration determine the trajectory. Understanding this concept is crucial for interpreting the height function and its inverse in the context of the baseball's flight.