Problem 43 - Algebra and Trigonometry with Analytic Geometry

Question

Guinness Book of World Records reports that German shepherds can make vertical leaps of over 10 feet when scaling walls. If the distance \(s\) (in feet) off the ground after \(t\) seconds is given by the equation \(s=-16 t^{2}+24 t+1\), for how many seconds is the dog more than 9 feet off the ground?

Step-by-Step Solution

Answer

The dog is more than 9 feet off the ground for 0.5 seconds, between 0.5 seconds and 1 second.

Step 1: Understand the Problem

We need to determine the time interval during which the height of the German shepherd is more than 9 feet above the ground. The height function is given as \(s(t) = -16t^2 + 24t + 1\). We need \(s(t) > 9\).

Step 2: Formulate the Inequality

Set up the inequality based on the condition that the dog should be more than 9 feet off the ground: \(-16t^2 + 24t + 1 > 9\).

Step 3: Simplify the Inequality

Subtract 9 from both sides to set the inequality in a standard form: \(-16t^2 + 24t + 1 - 9 > 0\), which simplifies to \(-16t^2 + 24t - 8 > 0\).

Step 4: Solve the Quadratic Inequality

First, solve the equation \(-16t^2 + 24t - 8 = 0\) to find the points where the height is exactly 9 feet. Use the quadratic formula: \[t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]For \(a = -16\), \(b = 24\), and \(c = -8\), calculate the determinant \(b^2 - 4ac = 576 - 512 = 64\).

Step 5: Calculate Roots

Calculate the roots using the quadratic formula:\[t = \frac{-24 \pm \sqrt{64}}{-32}\]The roots are \(t = \frac{-24 + 8}{-32} = \frac{-16}{-32} = 0.5\) and \(t = \frac{-24 - 8}{-32} = 1\).

Step 6: Determine the Interval

Since the parabola described by \(-16t^2 + 24t - 8\) opens downwards, the interval where the inequality holds (i.e., where the height \(s(t)\) is greater than 9 feet) is between the roots. Thus, the interval is \(0.5 < t < 1\).

Step 7: Conclusion

The German shepherd is more than 9 feet off the ground between \(0.5\) seconds and \(1\) second.

Key Concepts

Quadratic Equation

Quadratic equations are a fundamental concept in algebra and are typically written in the form \[ax^2 + bx + c = 0\]. The coefficients \(a\), \(b\), and \(c\) are constants, where \(a eq 0\). The quadratic equation is crucial for solving various mathematical problems, including those involving calculations of trajectory or motion, such as the vertical leap of a dog.
When solving a quadratic equation, you might use different methods such as:

• The quadratic formula: \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]
• Factoring, if the equation can be easily broken down into simpler binomial components
• Completing the square, which restructures the equation into a perfect square trinomial

Each method is useful in different scenarios, but the quadratic formula is the most versatile as it applies to all quadratic equations.

Parabola

The graph of a quadratic equation is called a parabola. Parabolas have several significant properties:

• The vertex, which is the highest or lowest point, depending on whether the parabola opens upwards or downwards.
• The axis of symmetry, a vertical line that divides the parabola into two mirror-image halves.
• The direction of opening, determined by the sign of \(a\) in the quadratic equation.

In our problem, the function \(s(t) = -16t^2 + 24t + 1\) represents a parabola that opens downwards because the coefficient \(a\) is negative.
Understanding the properties of parabolas helps in figuring out key intervals for solving inequalities, such as determining the time for which the height exceeds a certain value.

Inequality Solving

Inequalities represent a comparison between expressions and are solved to find a range of values for variables. Quadratic inequalities, like the one in our exercise, involve solving an inequality of the form \[ax^2 + bx + c > 0\] (or less than or equal to, depending on the scenario).
To solve our quadratic inequality, we first convert it to an equation \[ax^2 + bx + c = 0\] to find critical points or roots. These roots divide the number line into intervals which can then be tested to determine where the inequality holds true. In our case, solving \[-16t^2 + 24t - 8 = 0\] produced roots that helped establish the interval \[0.5 < t < 1\], where the height of the dog is more than 9 feet off the ground.

Vertical Motion

Vertical motion problems frequently use quadratic equations to describe how objects move over time under the influence of gravity. This involves concepts like:

• Initial velocity, which affects how quickly an object gains or loses height.
• Acceleration due to gravity, which is approximately \(-16\, \text{feet/second}^2\) for objects near the earth's surface when measured in feet.

In the problem of the German shepherd, the equation \(s(t) = -16t^2 + 24t + 1\) encapsulates these principles. Here:

• \(-16t^2\) accounts for the acceleration due to gravity
• \(24t\) represents the initial upward velocity
• The constant \(+1\) indicates the initial height from which the leap started

Understanding vertical motion through this equation helped us determine the intervals when the dog was above a given height during its leap.