Problem 19 - A Graphical Approach to Precalculus with Limits

Question

Solve each problem. 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 determine when the baseball hits the ground. The baseball's height is given by the equation \( s(t) = 75 - 16t^2 \), and it hits the ground when \( s(t) = 0 \). This means we need to solve the equation for \( t \) when \( 75 - 16t^2 = 0 \).

Step 2: Setting up the Equation

Start with the equation for the height of the baseball: \( 75 - 16t^2 = 0 \). This represents the condition where the baseball is at the ground level (height of 0 feet).

Step 3: Solving the Equation for Time

To find \( t \), rearrange the equation. Add \( 16t^2 \) to both sides to get \( 75 = 16t^2 \), then divide both sides by 16 to find \( t^2 \). The equation simplifies to \( t^2 = \frac{75}{16} \).

Step 4: Calculating the Time

Calculate \( t \) by taking the square root of both sides: \( t = \sqrt{\frac{75}{16}} \). Simplifying inside the square root gives \( t = \sqrt{4.6875} \), which approximately equals 2.165 seconds.

Step 5: Rounding the Result

Round 2.165 to the nearest tenth of a second. Therefore, \( t \approx 2.2 \) seconds.

Key Concepts

Projectile Motion

When an object is thrown or dropped in the air, it is affected by gravity, giving it a curved path known as projectile motion. This motion can be observed in many situations, like a baseball being dropped or thrown.
In the case of a baseball being dropped from a stadium seat, the only force acting on it is gravity. This force pulls it towards the ground, following a parabolic path. The equation describing this motion, namely \[s(t) = 75 - 16t^2,\] shows the height of the baseball above ground as a function of time, given in seconds.

• The number 75 is the initial height of the baseball.
• The term \(-16t^2\) represents the effect of gravity on the baseball. The constant \(-16\) comes from the gravitational acceleration, expressed in feet per second squared.

This parabolic path is a classic example of quadratic motion in physics. Understanding this helps in solving problems related to objects in free fall.

Solving by Factoring

Factoring is one of the methods used to solve quadratic equations. A quadratic equation is generally represented in the form \[ax^2 + bx + c = 0.\] However, not every quadratic equation can be solved directly by factoring, especially when it simplifies to a form like \[75 - 16t^2 = 0,\] from our exercise.
In most cases, you may need to rearrange the equation and potentially use other techniques if direct factoring isn't applicable.

Let's restate the problem: We started with \(75 - 16t^2 = 0\), which becomes \(t^2 = \frac{75}{16}\) after rearranging terms and ensuring that the equation is suitable for the application of the next solving method, like taking the square root.

• The factoring method would typically look for two numbers that multiply to \(ac\) and add to \(b\).
• When the quadratic cannot be easily factored, using a different strategy will often get you to the solution more efficiently.

Square Root Method

The square root method is a direct and efficient way to solve quadratic equations when they can be expressed in the form \[x^2 = k,\] where \(k\) is a constant. This method is particularly useful after manipulating the original quadratic equation to isolate the squared term.
From the example, we derived \(t^2 = \frac{75}{16}\). Applying the square root to both sides, we get: \[t = \sqrt{\frac{75}{16}}.\] By simplifying the square root, we approximate \(t\) to be around 2.165 seconds. Rounding this value, we find that the baseball takes about 2.2 seconds to hit the ground.

This method is handy because:

• It's quick when the equation is already in the form \(x^2 = k\).
• It avoids lengthy calculations and simplifies the solving process.
• It is straightforward and relies on basic algebraic manipulation.

Understanding and applying the square root method also builds foundational skills for more complex algebraic solutions.