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Magazine Chapter 9: Problem 9
The equation \(h=-4.9 t^{2}+17 t+2.2\) models the height of a stone thrown into the air, where \(t\) is in seconds and \(h\) is in meters. Use the quadratic formula to find how long the stone is in the air.
Short Answer
Expert verified
The stone is in the air for approximately 3.6 seconds.
Step by step solution
01
Identify Quadratic Components
The equation given is \(h=-4.9t^2+17t+2.2\). It can be compared to the general form of a quadratic equation, \(ax^2+bx+c=0\). Here, \(a = -4.9\), \(b = 17\), and \(c = 2.2\).
02
Use the Quadratic Formula
The quadratic formula is \(t = \frac{-b \pm \sqrt{b^2-4ac}}{2a}\). We'll use this formula to find \(t\) when \(h = 0\) (the stone hits the ground). Substitute \(a = -4.9\), \(b = 17\), and \(c = 2.2\).
03
Calculate the Discriminant
The discriminant \(b^2-4ac\) is calculated as follows: \(17^2 - 4(-4.9)(2.2) = 289 + 43.12 = 332.12\). Since the discriminant is positive, there will be two real solutions.
04
Compute the Solutions
Using the discriminant in the quadratic formula, the solutions for \(t\) are \(t = \frac{-17 \pm \sqrt{332.12}}{-9.8}\). Compute the square root and solve for both plus and minus cases.
05
Calculate the Square Root
The square root of the discriminant is \(\sqrt{332.12} \approx 18.22\).
06
Find Both Values of t
Substitute back into the quadratic formula to find both values: \(t_1 = \frac{-17 + 18.22}{-9.8} \approx -0.12\) (ignore this as time cannot be negative) and \(t_2 = \frac{-17 - 18.22}{-9.8} \approx 3.6\).
07
Conclude the Solution
The valid solution for the time the stone is in the air (from initial throw to when it hits the ground) is \(t \approx 3.6\) seconds.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Quadratic Formula
The quadratic formula is a powerful tool used to find the solutions of quadratic equations. It is expressed as \( x = \frac{-b \pm \sqrt{b^2-4ac}}{2a} \), where \( a \), \( b \), and \( c \) are the coefficients from the quadratic equation \( ax^2 + bx + c = 0 \). This formula allows you to calculate the values of \( x \) that satisfy the equation, giving you potential points where a parabola intersects the x-axis. To use this formula efficiently, ensure the equation is set to zero and coefficients are correctly identified.
Discriminant
The discriminant is a key component of the quadratic formula. It is the expression under the square root, given by \( b^2 - 4ac \). The discriminant tells us the nature of the roots of a quadratic equation:
- If it is positive, there are two distinct real solutions.
- If it equals zero, there is exactly one real solution, indicating that the parabola touches the x-axis at one point.
- If it is negative, there are no real solutions as the parabola does not intersect the x-axis at any point.
Solving Quadratic Equations
Solving quadratic equations involves finding the values of the variable that make the equation true. Quadratic equations are polynomials of degree two, often written in standard form as \( ax^2 + bx + c = 0 \). There are various methods to solve them:
- Factoring, which requires expressing the quadratic as a product of two binomials.
- Completing the Square, a method that involves transforming the equation into a perfect square trinomial.
- Quadratic Formula, which is reliable and can be used irrespective of whether the quadratic can be factored.
Projectile Motion
Projectile motion refers to the behavior of an object thrown into the air and subject to gravity. The path followed by the object is typically parabolic, described by a quadratic equation. For example, the quadratic equation \( h=-4.9t^2+17t+2.2 \) models the height of a stone over time, where \( h \) is height and \( t \) is time. In our scenario, using this equation, the stone鈥檚 time aloft is found by setting \( h \) to zero (when it hits the ground) and solving for \( t \). This showcases the use of quadratic equations to model real-world phenomena, like the flight path of a projectile.
Algebra
Algebra is the branch of mathematics dealing with symbols and the rules for manipulating them. It forms the foundation for understanding quadratic equations and utilizing the quadratic formula. In algebra, you learn to work with equations, variables, and various forms of expressions. Mastery of algebra is essential for solving equations, modeling situations like projectile motion, and understanding abstract math concepts. By applying these algebraic methods, complex problems can be simplified, enabling deeper insights into mathematical relationships and real-world situations.
