Problem 21 - Algebra A Combined Function

Question

A rocket is fired upward from the ground with an initial velocity of 200 feet per second. Neglecting air resistance, the height of the rocket at any time t can be described in feet by the polynomial \(-16 t^{2}+200 t\). Find the height of the rocket at the time given in Exercises 19 through 22. See Example 5. $$ \begin{array}{c|c} \text { Time, } \boldsymbol{t} & \text { Height } \\ \text { (in seconds) } & -16 t^{2}+200 t \\ \hline 7.6 & \\ \hline \end{array} $$

Step-by-Step Solution

Answer

At \( t = 7.6 \) seconds, the rocket's height is 595.84 feet.

Step 1: Understand the Formula

The height of the rocket as a function of time, \( t \), is given by the polynomial \(-16t^2 + 200t\). This formula accounts for the initial upward velocity and the effect of gravity.

Step 2: Substitute the Time Value

We need to find the height at \( t = 7.6 \) seconds. Substitute 7.6 into the equation: \(-16(7.6)^2 + 200(7.6)\).

Step 3: Calculate \( t^2 \)

First, calculate \( 7.6^2 \). This is equal to 57.76.

Step 4: Compute \(-16t^2\)

Substitute \( t^2 = 57.76 \) into \(-16t^2\): \(-16 \times 57.76 = -924.16\).

Step 5: Compute \(200t\)

Calculate \(200 \times 7.6 = 1520\).

Step 6: Sum the Results

Add the results from steps 4 and 5: \(-924.16 + 1520 = 595.84\). This is the height of the rocket at \( t = 7.6 \) seconds.

Key Concepts

Polynomial Equations

Polynomial equations play a crucial role in mathematics and in modeling real-world phenomena. A polynomial equation consists of terms that can involve variables raised to whole-number exponents, and sometimes a constant term. In this context, the rocket follows the polynomial function based on projectile motion.

• The standard form of a quadratic polynomial is expressed as \( ax^2 + bx + c \), where \( a, b, \) and \( c \) are constants.
• In our rocket's height formula, \(-16t^2 + 200t\), the term \(-16t^2\) represents the effect of gravity pulling the rocket downwards, and \(200t\) is the initial upward velocity contributing to height.
• The polynomial helps predict the rocket's height at any given time, \( t \), by substituting values into the equation.

Understanding polynomial equations is valuable as they extend beyond simple computations. They help describe relationships and changes, which is essential in fields like physics, engineering, and economics.

Projectile Motion

Projectile motion, such as the path of a rocket, involves factors like initial velocity and gravity. This motion typically follows a parabolic trajectory, meaning that the path forms a curve shaped like a parabola.

• In our specific case, the initial velocity of the rocket is 200 feet per second. This is crucial as it defines how high and how fast the object will move.
• The polynomial formula \(-16t^2 + 200t\) models the vertical motion of the rocket, where the highest point (or apex) occurs when the vertical velocity is zero.

The formula serves to calculate the rocket's height at any point. For example, if you want to find out how high the rocket is after several seconds, you simply plug the time into the polynomial. By understanding these principles, you gain insight into how objects move under the influence of forces.

Gravity Effects

Gravity is the force that constantly pulls objects toward the center of the Earth. It affects all objects in projectile motion, such as a rocket launched into the sky. In our equation, the effect of gravity is represented by the term \(-16t^2\).

• The value \(-16\) comes from physics, where the acceleration due to gravity is approximately \(-32 \text{ ft/s}^2\) on Earth, and since this is a formula considering the vertical component, it's halved.
• Gravity's negative sign indicates that it pulls back on the upward motion, decelerating the rocket until it reaches its peak and begins descending.

Understanding gravity's role helps explain why our rocket will not rise indefinitely. After reaching its maximum height, gravity causes the rocket to fall back toward the ground. The study of its effects allows scientists and engineers to design safer and more efficient technologies related to motion and flight.