Problem 44 - Introductory Algebra for College Students

Question

A football is kicked straight up from a height of 4 feet with an initial speed of 60 feet per second. The formula $$h=-16 t^{2}+60 t+4$$ describes the ball's height above the ground, \(h\), in feet, \(t\) seconds after it is kicked. Will the ball reach a height of 80 feet? Substitute 80 for \(h\) in the given formula and solve the equation. Are the solutions real numbers? Explain why the ball will or will not reach 80 feet.

Step-by-Step Solution

Answer

Yes, the ball will reach 80 feet 6.125 seconds after being kicked.

Step 1: Substitute 80 for h

We substitute 80 for \(h\) in the equation \(h=-16 t^{2}+60 t+4\) to get the equation \(-16 t^{2}+60 t+4 - 80 = 0\). This simplifies to \(-16 t^{2}+60 t-76 = 0.\)

Step 2: Solve the quadratic equation

We use the quadratic formula \(t = \frac{-b \pm \sqrt{b^{2}-4ac}}{2a}\) to solve for \(t\). Here \(a = -16\), \(b = 60\) and \(c = -76\). Thus, \(t = \frac{-60 \pm \sqrt{(60)^{2}-4*(-16)*(-76)}}{2*(-16)}\). This simplifies to \(t=\frac{-60\pm\sqrt{3600-(-1216)}}{-32} = \frac{-60\pm\sqrt{4816}}{-32}=\frac{-60\pm 4*34}{-32}\). We obtain two roots: \(t = \frac{-60 + 136}{-32} = -2.375\) (which doesn't make sense in this context as time cannot be negative), and \(t = \frac{-60 - 136}{-32}=6.125\), which is realistic.

Step 3: Conclude whether the ball reaches 80 feet

Since one of the answers for \(t\) is a real positive number, the ball will reach a height of 80 feet in 6.125 seconds. The negative answer indicates the time before the ball was kicked, this doesn't provide any useful information in this context.

Key Concepts

Height of a Projectile

Understanding the height of a projectile is crucial in predicting the motion of objects like a football when they are thrown or kicked upwards. The height of a projectile changes over time due to the forces of gravity and the initial force applied. In this scenario, a football is kicked upwards and follows a path determined by a quadratic equation:\[ h = -16t^2 + 60t + 4 \]This formula provides the height \( h \) at any time \( t \) after the football is kicked. The terms in this equation represent:

• \(-16t^2\): The effect of gravity pulling the projectile downwards.
• \(60t\): How the initial kick affects the height, working against gravity initially.
• 4: The starting height of the kick.

This straightforward formula can tell us how high the ball will go at various times, helping determine if it can reach a certain height like 80 feet.

Initial Velocity

The initial velocity is a key component in determining the trajectory of a projectile. In this exercise, the football is given an initial velocity of 60 feet per second, which represents the speed and direction of the ball at the moment it is kicked. This initial force propels the football upwards, countering gravity temporarily.Initial velocity can drastically affect:

• How high the projectile will rise.
• How far it will travel, if also moving horizontally.

For a vertical kick like the football in the exercise, the higher the initial velocity, the higher the ball will reach. The quadratic equation to describe the ball's height includes this velocity as a critical factor, influencing the term \(60t\). This term represents how much the football climbs due to its initial kick before gravity slows it down.

Solving Quadratic Equations

Quadratic equations come into play when describing the motion of projectiles, making them essential in physics and engineering. A quadratic equation has the form:\[ ax^2 + bx + c = 0 \]To solve this equation, we often use the quadratic formula:\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]In our scenario with the football, after substituting 80 for the height \( h \), the equation becomes:\[ -16t^2 + 60t - 76 = 0 \]By using the quadratic formula and the values \( a = -16 \), \( b = 60 \), and \( c = -76 \), we can find values of \( t \). These values indicate the time at which the football reaches a given height, in this case, deciding whether it reaches 80 feet.

Real Solutions

In mathematics, real solutions refer to the actual, plausible numbers that solve an equation. For a quadratic equation derived from a physical scenario like the height of a projectile, we are primarily interested in positive real solutions. These solutions indicate feasible times after an event, like a football being kicked.When we solve the equation:\[ -16t^2 + 60t - 76 = 0 \]Two solutions for \( t \) emerge: one is negative, \(-2.375\), and cannot represent a real-world scenario since negative time doesn't make sense in this context. The positive solution, \(6.125\), is physically meaningful and denotes the time in seconds after the kick when the ball reaches 80 feet. This means the football will indeed reach that height at \(6.125\) seconds, verifying the reachability of the desired height.