Problem 22 - 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. $$ \begin{array}{c|c} \text { Time, } \boldsymbol{t} & \text { Height } \\ \text { (in seconds) } & -16 t^{2}+200 t \\ \hline 10.3 & \\ \hline \end{array} $$

Step-by-Step Solution

Answer

The height at 10.3 seconds is 362.56 feet.

Step 1: Understand the Equation

The equation given is a quadratic function describing the height of a rocket over time. The formula is \( h(t) = -16t^2 + 200t \), where \( h \) is the height in feet and \( t \) is time in seconds.

Step 2: Substitute Time Value

To find the height at a specific time, substitute \( t = 10.3 \) into the equation. Replace \( t \) with 10.3 in the equation \( h(t) = -16t^2 + 200t \) to get \( h(10.3) = -16(10.3)^2 + 200(10.3) \).

Step 3: Calculate \(-16t^2\)

Calculate \(-16(10.3)^2\). First, compute \( (10.3)^2 \), which is 106.09. Then multiply by -16 to get \(-16 \times 106.09 = -1697.44\).

Step 4: Calculate \(200t\)

Calculate \(200 \times 10.3\) which equals 2060.

Step 5: Add Results

Combine the results from Step 3 and Step 4 to find the height \( h(10.3) \). Add \(-1697.44 + 2060 = 362.56\).

Step 6: Conclusion

The height of the rocket at \( t = 10.3 \) seconds is 362.56 feet.

Key Concepts

Polynomial Equations

Polynomial equations involve expressions that include variables raised to whole number exponents. They play a crucial role in mathematics as they help in modeling a variety of real-world situations. A polynomial in one variable, say \( t \), is an expression like \( a_n t^n + a_{n-1} t^{n-1} + \ldots + a_1 t + a_0 \) where \( a_0, a_1, \ldots, a_n \) are constants, and \( n \) is a non-negative integer. The equation \( h(t) = -16t^2 + 200t \) given in the exercise is a quadratic polynomial because it includes both \( t^2 \) and \( t \) terms. Quadratic polynomials are specifically important in physics for modeling projectile motion. Understanding the structure of polynomials helps in solving them by substitution, factoring, or using quadratic formulas.

Projectile Motion

Projectile motion refers to the motion of an object that is propelled into the air and subject to gravitational force. In the case of the rocket in our exercise, the motion is modeled perfectly by a quadratic function.The function \( h(t) = -16t^2 + 200t \) describes a downward-opening parabola. The coefficient \(-16\) stems from the acceleration due to gravity (in feet per second squared), while the \(200t\) term represents the initial upward velocity in feet per second. Key points to analyze:

• The initial velocity when \( t = 0 \) is equal to 200 feet per second.
• The negative coefficient of \( t^2 \) reflects the deceleration due to gravity.
• At different times \( t \), substituting \( t \) into the equation provides the rocket's height at that specific moment.

Understanding projectile motion through equations like this helps in various fields such as engineering, sports, and aerospace.

Substitution Method

The substitution method involves replacing a variable with a numerical value to solve equations easily. This method is integral in solving polynomial equations, especially when you need to determine specific outcomes.In the exercise, to find out the rocket's height at \( t = 10.3 \) seconds, we substitute \( t \) with 10.3 into the equation \( h(t) = -16t^2 + 200t \). By doing so, we transform an abstract function into a calculable task:

• First, compute \( (10.3)^2 \) to get 106.09.
• Multiply by -16 giving \(-1697.44\).
• Calculate \( 200 \times 10.3 = 2060 \).
• Add the results: \(-1697.44 + 2060 = 362.56 \).

This technique ensures you can plug in different values of \( t \) and easily find corresponding heights, making it especially useful for real-time problem-solving in physics.

Physics in Algebra

Algebra is not just about solving equations in a vacuum; it often intersects with physics to solve real-world problems. This intersection is a powerful tool, allowing us to model and predict physical phenomena using mathematical equations.When analyzing projectile motion like in the exercise, algebra allows us to understand how variables interact. The equation \( h(t) = -16t^2 + 200t \) describes how the rocket's height changes based on time due to physical forces like gravity. Benefits of understanding physics through algebra:

• It provides precise models for predicting outcomes in engineering and space travel.
• Makes complex ideas accessible, translating real-world movement into a mathematical framework.
• Improves problem-solving skills by linking abstract algebraic concepts with tangible results.

Mastering how physics utilizes algebraic equations equips students with crucial skills for fields that demand precision and analytical thinking.