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

Lial13th EditionCollege AlgebraISBN: 9780136881063Not the one you use?Change textbook

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Lial 13th Edition

Ch. 1 - Equations and Inequalities

Problem 94

Chapter 2, Problem 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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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.