Problem 101 - College Algebra with Modeling and Visualization

Question

A baseball is dropped from a stadium seat that is 75 feet above the ground. Its height \(s\) in feet after \(t\) seconds is given by \(s(t)=75-16 t^{2} .\) Estimate to the nearest tenth of a second how long it takes for the baseball to strike the ground.

Step-by-Step Solution

Answer

The baseball takes approximately 2.2 seconds to hit the ground.

Step 1: Understanding the Problem

We need to find the time it takes for the baseball to hit the ground. This happens when the height of the baseball, given by the function \(s(t) = 75 - 16t^2\), is equal to 0.

Step 2: Setting Up the Equation

To find when the baseball hits the ground, set the height equation equal to 0: \(75 - 16t^2 = 0\). Solve this equation for \(t\).

Step 3: Solving the Equation

Rearrange the equation to isolate \(t\): \(16t^2 = 75\). Then divide both sides by 16: \(t^2 = \frac{75}{16}\).

Step 4: Taking the Square Root

Take the square root of both sides to solve for \(t\): \(t = \sqrt{\frac{75}{16}}\). Simplify it further.

Step 5: Calculating the Final Answer

Calculate \(\sqrt{\frac{75}{16}}\) to find \(t\). This yields \(t \approx 2.17\). Thus, the time taken to the nearest tenth of a second is 2.2 seconds.

Key Concepts

Projectile Motion

Projectile motion describes the motion of an object dropped or thrown and subjected only to the force of gravity. In this baseball problem, when it is dropped, it moves in a vertical path downwards following a parabolic trajectory. The motion is governed by the quadratic equation \(s(t) = 75 - 16t^2\), which reflects that gravity continuously accelerates the baseball downwards.

• The initial height is 75 feet, indicating where it starts falling.
• The term \(-16t^2\) accounts for the acceleration due to gravity trying to pull the baseball down to the earth.

The equation ensures the baseball's height becomes zero at a certain point, marking when the ball has struck the ground. Understanding this type of motion is crucial as it helps predict where and when the baseball, or any object under similar conditions, will reach the ground.

Square Roots

Square roots are mathematical expressions used to find a number which, when multiplied by itself, yields the original number. Within the context of this problem, the use of square roots stems from solving the quadratic equation \(16t^2 = 75\). To isolate \(t\), the equation is manipulated as follows. 1. Steps to finding square roots include rearranging to \(t^2 = \frac{75}{16}\), thereby expressing \(t\) more clearly.2. To ascertain the value of \(t\), take the square root of both sides: \(t = \sqrt{\frac{75}{16}}\).This step is based on the principle that both positive and negative roots exist. However, in our scenario, only the positive root is meaningful since time cannot be negative, leading to a positive result which is approximately 2.17 seconds. Thus, the square root helps in simplifying and understanding real-world solutions.

Problem Solving Steps

Problem-solving involves sequential processes to address and answer real-world problems, like the time it takes for a baseball to hit the ground. Let's break it down as done in our solution. 1. **Understand the Problem**: Know what is required—calculate when the baseball hits the ground using the height equation.2. **Equation Setup**: Match the height equation \(s(t)=75 - 16t^2\) to zero to determine this instance.3. **Rearrange and Solve the Equation**: Solve for \(t\) by manipulating the equation to \(16t^2 = 75\) and further to \(t^2 = \frac{75}{16}\).4. **Use Square Roots**: Apply square roots to solve \(t\), which simplifies understanding and solves for the exact time.Each step refines the focus on the problem and ensures a logical conclusion is drawn. Through structured problem-solving methods, real-world issues involving math can be navigated effectively, yielding precise results.