Solve each problem. Velocity of an Object The velocity of an object, v, after t seconds is given by v=3t2-18t+24.Find the interval where the velocity is negative.

Start with the given velocity function: $$v = 3t$$^{2}$$ - 18t + 24. To $$find where the velocity is negative, set up the inequality: $$3t^{2} - 18t + 24$$ \u003c 0$$. Divide the entire inequality by 3 to simplify: t$$^{2}$$ - 6t + 8 \u003c 0. $$Factor the quadratic expression: $$(t - 2)(t - 4) \u003c 0. $$Determine the intervals where the product $$(t - 2)(t - 4) is $$less than zero by analyzing the sign of each factor on the number line.

Ch. 1 - Equations and Inequalities

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Lial 13th Edition

Ch. 1 - Equations and Inequalities

문제 94

2장, 문제 94

Solve each problem. Velocity of an Object The velocity of an object, v, after t seconds is given by v=3t²-18t+24.Find the interval where the velocity is negative.

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Start with the given velocity function: $v = 3t^{2} - 18t + 24$.

To find where the velocity is negative, set up the inequality: $3t^{2} - 18t + 24 < 0$.

Divide the entire inequality by 3 to simplify: $t^{2} - 6t + 8 < 0$.

Factor the quadratic expression: $(t - 2)(t - 4) < 0$.

Determine the intervals where the product $(t - 2)(t - 4)$ is less than zero by analyzing the sign of each factor on the number line.

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주요 개념

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

Quadratic Functions

A quadratic function is a polynomial of degree two, generally written as ax² + bx + c. Its graph is a parabola, which can open upwards or downwards depending on the sign of 'a'. Understanding the shape and properties of quadratics helps analyze the behavior of the velocity function over time.

Finding Roots of a Quadratic Equation

Roots or zeros of a quadratic function are the values of the variable that make the function equal to zero. These can be found using factoring, completing the square, or the quadratic formula. Identifying roots is essential to determine intervals where the function changes sign.

Sign Analysis of Functions

Sign analysis involves determining where a function is positive, negative, or zero by testing values in intervals defined by its roots. For velocity, this helps find when the object moves forward (positive velocity) or backward (negative velocity) by checking the sign of v(t) between roots.