Problem 132 - College Algebra

Question

A person throws a rock upward from the edge of an 80 -foot cliff. The height, \(h,\) in feet, of the rock above the water at the bottom of the cliff after \(t\) seconds is described by the formula $$h=-16 t^{2}+64 t+80$$ How long will it take for the rock to reach the water?

Step-by-Step Solution

Answer

The rock will hit the water after the amount of time found as the positive root of the quadratic equation.

Step 1: Identifying the problem

First, we need to understand that we are dealing with a quadratic equation in the form \( h = -16t^2 + 64t + 80 \), which describes the motion of a rock thrown upward from a cliff. We are asked to find the time it will take for the rock to reach the water. This means we have to find when the height \( h \) of the rock is equal to zero.

Step 2: Setting the equation

To find the time at which the rock will hit the water, we must set the height \( h \) to be equal to zero, since when the rock hits the water it will be at height zero relative to the water. This results in the equation \( 0 = -16t^2 + 64t + 80 \).

Step 3: Solving the equation

To find the solution, we can arrange the equation and solve it using the quadratic formula. The quadratic formula that is used to solve equations in the form \( ax^2 + bx + c = 0 \) is \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). Using this formula, we can calculate the values of \( t \) that will make \( h = 0 \). These values will be the durations for which the rock is in the air.

Step 4: Rejecting negative solutions

After solving the quadratic equation, two solutions will be found, \( t_1 \) and \( t_2 \). Time \( t \) should always be a positive value since it doesn't make sense to have a negative time duration in this context. Therefore, we discard the negative value, meaning the correct \( t \) is the positive solution.

Key Concepts

Understanding Rock Trajectory

When a rock is thrown upward from a cliff, it follows a specific path due to the forces acting on it. This path is known as the rock's trajectory. The trajectory is affected by gravity, which pulls the rock downward, and the initial velocity, which launches the rock upward. In our problem, the trajectory is described by the quadratic equation \( h = -16t^2 + 64t + 80 \). Here, \( h \) represents the height of the rock above the water over time \( t \). The negative coefficient of \( t^2 \) tells us that the rock's path will be a parabola opening downwards. This means the rock will first rise, reach a maximum height, and then descend until it eventually hits the water at the bottom of the cliff.

Calculating Time to Impact

Time to impact refers to the duration from when the rock is thrown until it hits the water. To find out this time, we set the height \( h \) equal to zero because once the rock hits the water, it won't be above it anymore. This gives us the equation \( 0 = -16t^2 + 64t + 80 \). Solving this equation for \( t \) will provide the possible time values when the rock reaches the water. It's crucial to note that when calculating time, we only consider positive values. Time cannot be negative, so we disregard any negative solutions that might arise from solving the equation.

Using the Quadratic Formula

The quadratic formula is a powerful tool used to solve quadratic equations, especially when they don't factor easily. The standard form for a quadratic equation is \( ax^2 + bx + c = 0 \), and the quadratic formula is given by \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \].To apply this formula to our problem, we set \( a = -16 \), \( b = 64 \), and \( c = 80 \). Plugging these values into the formula helps us find the values of \( t \), which indicate how long the rock will stay above the water. The formula will yield two possible solutions, but only the positive one is physically meaningful in this context, as time cannot be negative.

Applying Motion Equations

Motion equations are fundamental principles used to describe the behavior of objects in motion. For this exercise, the focus is on a specific quadratic motion equation that models the rock's flight. The equation \( h = -16t^2 + 64t + 80 \) stems from basic principles of physics, particularly kinematics.

• -16t²: This term represents the influence of gravity, which consistently pulls the rock downwards, decelerating its initial upward velocity.
• 64t: This is the initial velocity component, showing how powerfully the rock is thrown upward.
• +80: This is the initial height, accounting for the rock already being 80 feet above the water due to the cliff's height.

Understanding these components enables us to make accurate predictions about the rock's motion and solve for the time it will impact the water. This applied blend of physics and mathematics helps us navigate real-world problems effectively.