Problem 59 - Algebra 2 and Trigonometry

Question

A soccer ball is kicked upward from ground level with an initial velocity of 52 feet per second. The equation \(\mathrm{h}(t)=-16 t^{2}+52 t\) gives the ball's height in feet after \(t\) seconds. To the nearest tenth of a second, during what period of time was the height of the ball at least 20 feet?

Step-by-Step Solution

Answer

The ball's height is at least 20 feet from \(t \approx 0.4\) to \(t \approx 3.1\) seconds.

Step 1: Understand the problem

We need to find the time intervals when the height of the ball is at least 20 feet. This means solving the inequality \(-16t^2 + 52t \geq 20\).

Step 2: Set up the inequality

Substitute 20 into the height equation: \(-16t^2 + 52t = 20\). Rearrange the equation to \(-16t^2 + 52t - 20 \geq 0\).

Step 3: Solve the quadratic equation

Rearrange the inequality as an equation: \(-16t^2 + 52t - 20 = 0\). Now use the quadratic formula: \(t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a = -16\), \(b = 52\), and \(c = -20\).

Step 4: Calculate discriminant

Calculate \(b^2 - 4ac\): \(52^2 - 4(-16)(-20) = 2704 - 1280 = 1424\).

Step 5: Find roots using quadratic formula

Substitute into the quadratic formula: \(t = \frac{-52 \pm \sqrt{1424}}{2(-16)}\) to find the roots. Calculate \(\sqrt{1424} \approx 37.7\).

Step 6: Calculate exact times

Evaluate \(t = \frac{-52 + 37.7}{-32}\) and \(t = \frac{-52 - 37.7}{-32}\), resulting in two solutions: \(t \approx 0.4\) seconds and \(t \approx 3.1\) seconds.

Step 7: Interpret the solution

The ball's height is at least 20 feet between \(t \approx 0.4\) seconds and \(t \approx 3.1\) seconds.

Key Concepts

Quadratic Formula

The quadratic formula is a fundamental tool in algebra, primarily used to solve quadratic equations. These are equations of the form \(ax^2 + bx + c = 0\). The quadratic formula helps find the values of \(x\) that satisfy the equation. It is represented as: \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\] Where:

• \(a\), \(b\), and \(c\) are coefficients in the quadratic equation.
• \(b^2 - 4ac\) is known as the discriminant.

The discriminant determines the nature of the roots, which can be real and distinct, real and repeated, or complex. As you can see, the quadratic formula is indispensable for finding the exact roots of any quadratic equation.

Inequality Solving

Solving inequalities involves finding the set of values that satisfy the inequality condition. In our example, we solved the inequality \(-16t^2 + 52t \geq 20\) to determine when the soccer ball's height is at least 20 feet. Here's how you might approach solving such an inequality:

• First, rewrite the inequality in standard form: \(-16t^2 + 52t - 20 \geq 0\).
• Next, treat the inequality as an equation: \(-16t^2 + 52t - 20 = 0\) and solve for \(t\) using the quadratic formula.
• Use these solutions to determine the intervals to test in the inequality.
• Check each interval by substituting back into the original inequality to see which intervals satisfy the inequality.

This method ensures you find the set of \(t\) values for which the inequality holds true.

Physics in Mathematics

Mathematics often serves as the language of physics, enabling us to describe and analyze physical phenomena precisely. In our case, we use a quadratic equation, a common mathematical model, to comprehend the motion of a projectile, such as a soccer ball. Physics leverages mathematical functions to:

• Model motions, like the parabolic path of a projectile.
• Predict future states from current conditions, using equations of motion.
• Analyze forces acting on bodies, in our case, gravity acting on the soccer ball.

Understanding how mathematics models physics empowers us to solve complex real-world problems by leveraging abstract mathematical concepts.

Projectile Motion

Projectile motion is a type of motion experienced by an object that is projected into the air. It exhibits both horizontal and vertical motions, under the effect of gravity. In our exercise, the soccer ball's path represents a projectile motion. Key characteristics of projectile motion include:

• The trajectory is a parabola due to the uniform acceleration of gravity.
• It involves two velocity components: horizontal constant and vertical changing due to gravity.
• The maximum height, time of flight, and range can be analyzed using kinematic equations.

In mathematical terms, the height equation \(h(t) = -16t^2 + 52t\) reflects the vertical component, while gravity causes the quadratic nature of motion. Solving for the time the ball is above a certain height illustrates how mathematics can predict the very nuances of projectile paths.