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Chapter 5: Problem 15

A rock is thrown downward, and the distance \(D\), in feet, that it falls in \(t\) seconds is given by \(D=16 t^{2}+3 t\). Find how long it takes for the rock to fall 400 feet by using a. the quadratic formula. b. the crossing-graphs method.

Short Answer

Expert verified
The rock takes approximately 4.906 seconds to fall 400 feet.

Step by step solution

01

Reformulate the Equation

Start with the given equation for distance: \( D = 16t^2 + 3t \). Set \( D = 400 \) to find the time \( t \). This gives us the equation: \( 16t^2 + 3t = 400 \). Rearrange this to the standard quadratic form: \( 16t^2 + 3t - 400 = 0 \).
02

Identify Coefficients for the Quadratic Formula

The quadratic equation is \( 16t^2 + 3t - 400 = 0 \). The quadratic formula is \( t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). Identify \( a = 16 \), \( b = 3 \), and \( c = -400 \).
03

Calculate the Discriminant

Substitute \( a \), \( b \), and \( c \) into the discriminant formula: \( b^2 - 4ac \). Calculate: \( 3^2 - 4 \times 16 \times (-400) = 9 + 25600 = 25609 \).
04

Apply the Quadratic Formula

Substitute the values into the quadratic formula: \( t = \frac{-3 \pm \sqrt{25609}}{32} \). Calculate \( \sqrt{25609} = 160.06 \) approximately. Substitute this back into the equation: \( t = \frac{-3 \pm 160.06}{32} \).
05

Determine Valid Solution for Time

Calculate the two potential solutions: \( t = \frac{-3 + 160.06}{32} \) and \( t = \frac{-3 - 160.06}{32} \). Solving gives \( t = 4.906 \) and \( t \approx -5.094 \). Since time cannot be negative, the solution is \( t = 4.906 \) seconds.
06

Crossing-Graphs Method Setup

To use the crossing-graphs method, graph the equation \( y = 16t^2 + 3t \) and the horizontal line \( y = 400 \). Find the intersection points of these graphs.
07

Plotting and Finding Intersection

Using graphing tools, plot the graphs. The parabola \( y = 16t^2 + 3t \) and line \( y = 400 \) intersect at \( t \approx 4.906 \). This confirms our earlier result from the quadratic formula.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Graphing Methods
Graphing is a visual way to solve quadratic equations. In this method, you plot the given equation and a line on a graph. For our example of a rock falling, the equation we use is \( y = 16t^2 + 3t \), representing a parabola. To find out when the rock hits 400 feet, we also plot the horizontal line \( y = 400 \).
  • The x-axis typically represents time \( t \), while the y-axis is for the distance \( D \) that the rock falls.
  • Where the parabola intersects with the line \( y = 400 \), you find the time \( t \) that solves the equation \( 16t^2 + 3t = 400 \).
This intersection point visually indicates the solution. Graphing is especially useful when you are looking for an estimate or need to double-check algebraic solutions.
Discriminant
The discriminant is a key component of the quadratic formula that helps you determine the nature of the roots of a quadratic equation, noted as \( b^2 - 4ac \). In our rock-throwing problem, the quadratic is \( 16t^2 + 3t - 400 = 0 \). After identifying \( a = 16 \), \( b = 3 \), and \( c = -400 \), the discriminant is calculated as \( 9 + 25600 = 25609 \).
  • If the discriminant is positive, like in this case, it ensures there are two real and distinct solutions for \( t \).
  • A zero discriminant means there's exactly one real solution.
  • A negative discriminant signifies no real solution, implying imaginary roots.
Understanding the discriminant is crucial for knowing what type of solutions to expect from a quadratic equation.
Quadratic Formula
The quadratic formula is a universal method for solving any quadratic equation of the form \( ax^2 + bx + c = 0 \). It is written as \( t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). In the given problem, we used the quadratic formula to find the time it takes for the rock to fall 400 feet.First, identify your coefficients from the equation \( 16t^2 + 3t - 400 = 0 \); here, \( a = 16 \), \( b = 3 \), and \( c = -400 \). Substituting these into the formula gives:\[t = \frac{-3 \pm \sqrt{25609}}{32}\]After calculating \( \sqrt{25609} \approx 160.06 \), you end up with two potential solutions \( t = 4.906 \) and \( t = -5.094 \). Since time cannot be negative, \( t = 4.906 \) seconds is the valid solution. This method is precise and works every time there is a quadratic equation.
Intersection Points
In math and geometry, finding intersection points means locating where two graphs meet. For our rock problem, the intersection is between a parabola and a horizontal line.
  • The equation of the parabola is \( y = 16t^2 + 3t \).
  • The horizontal line is \( y = 400 \), representing the distance the rock needs to fall.
You determine where these two meet to find the value of \( t \) representing how long it takes the rock to reach 400 feet. Using a graphing tool or algebraic methods, these intersection points are found, providing not just the answer we need but confirming the accuracy of our problem solution. Intersection points offer a clear visual confirmation for when algebraic solutions might seem complex.

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