Problem 45 - Precalculus

Question

Involve vertical motion and the effect of gravity on an object. Because of gravity, an object that is projected upward will cventually reach a maximum height and then fall to the ground. The equation that determines the height \(h\) of a projectile \(t\) seconds after it is shot upward is given by $$ h=\frac{1}{2} a t^{2}+v_{0} t+h_{0} $$ where \(a\) is the acceleration due to gravity, \(h_{0}\) is the initial height of the object at time \(t=0,\) and \(v_{0}\) is the initial velocity of the object at time \(t=0 .\) Note that a projectile follows the path of a parabola opening down, so \(a<0\) A factory manufactures three types of golf balls: Eagle, Birdie, and Bogey. The daily production is 10,000 balls. The number of Eagle and Birdie balls combined equals the number of Bogey balls produced. If the factory makes three times more Birdie than Eagle balls, find the daily production of each type of ball.

Step-by-Step Solution

Answer

1,250 Eagle balls, 3,750 Birdie balls, and 5,000 Bogey balls are produced daily.

Step 1: Define Variables

Let's define the variables for each type of ball: let \( E \) be the number of Eagle balls, \( B \) be the number of Birdie balls, and \( G \) be the number of Bogey balls produced daily.

Step 2: Set Up Equations

Based on the problem, the following equations can be formulated:1. The total production is the sum of all types, so \( E + B + G = 10,000 \).2. The number of Eagle and Birdie balls combined equals the number of Bogey balls, so \( E + B = G \).3. The factory produces three times more Birdie balls than Eagle balls, so \( B = 3E \).

Step 3: Substitute and Simplify

Substitute equation 3, \( B = 3E \), into equation 2. This gives us \( E + 3E = G \) or \( G = 4E \). Substitute \( B = 3E \) and \( G = 4E \) into equation 1: \[ E + 3E + 4E = 10,000 \] \[ 8E = 10,000 \].

Step 4: Solve for Eagle Balls

Divide both sides of the equation \( 8E = 10,000 \) by 8 to solve for \( E \):\[ E = \frac{10,000}{8} \]\[ E = 1,250 \].

Step 5: Calculate Other Types

Use \( E = 1,250 \) to find the number of Birdie and Bogey balls:\[ B = 3E = 3 \times 1,250 = 3,750 \]\[ G = 4E = 4 \times 1,250 = 5,000 \].

Key Concepts

Projectile Motion

Projectile motion is a common phenomenon that occurs when an object is thrown or launched into the air and affected primarily by gravity. It follows a curved trajectory called a parabola. Let's break down the essential parts of understanding projectile motion:

• Trajectory: The path that the projectile follows is curved, and the shape is a parabolic arc. This shape occurs because of the influence of gravity, which continuously pulls the object downward.
• Initial Velocity: Often denoted as \( v_0 \), the initial velocity is the speed and direction at which the projectile is launched. It's crucial in determining how far and high the projectile will travel.
• Maximum Height: This is the peak point of the parabolic path where the object momentarily stops rising and begins to fall.
• Time of Flight: This is the total time the projectile remains in the air. It depends on the initial velocity and the angle of projection.

Understanding projectile motion involves analyzing these components to predict the projectile's behavior over time.

Gravity Effects

Gravity is a fundamental force that affects all objects with mass. For projectiles, gravity plays a pivotal role in determining their motion. Let's explore how gravity and its effects are essential in projectile motion:

• Acceleration Due to Gravity: Represented by \( a \) in the equation, is approximately \(-9.8 \text{ m/s}^2\) on Earth. This negative sign reflects that gravity pulls objects towards the planet's center, opposing their upward motion initially.
• Downward Force: Gravity constantly acts to bring the projectile back to the ground while influencing the path the object takes in the air (its trajectory).
• Effect on Motion: By consistently decelerating upwards motion and accelerating downwards motion, gravity ensures that the trajectory is a symmetrical parabola.

Understanding gravity is crucial for accurately calculating the maximum height and total distance traveled by a projectile.

Systems of Equations

Systems of equations often feature multiple equations that you need to solve simultaneously to find the unknown variables. In our example, we had a system that helped determine the production numbers for different types of golf balls.

Key steps in solving such systems include:

• Define Variables: Start by choosing variables to represent the unknowns—in this case, Eagle, Birdie, and Bogey balls.
• Write Equations: Convert word problems into algebraic equations. Here, three equations were derived from the problem description.
• Substitution or Elimination: Use methods such as substitution—as done by replacing variables with known equivalent expressions—or elimination to simplify and solve the equations.
• Find Solutions: Solving systematically finds values for each variable, providing practical, real-world applications for the system.

Systems of equations are powerful tools for organizing and solving complex problems efficiently.

Polynomial Functions

Polynomial functions are mathematical expressions involving variables raised to powers. They play a significant role in modeling real-life phenomena like projectile motion. Here's what you need to know about polynomial functions:

• Structure: These functions consist of terms that include variables with whole number exponents. For example, the height of a projectile over time is often represented by a quadratic polynomial.
• Degrees and Coefficients: The degree of the polynomial is the highest exponent, and the coefficients are the numbers in front of the variables. These determine the shape and position of the graph.
• Quadratic Functions: A special type of polynomial function with the highest degree of 2. They predict the parabolic path of projectiles, as demonstrated in the equation for projectile height \(h = \frac{1}{2} a t^2 + v_0 t + h_0\).
• Graphs: The graph of a polynomial can curve multiple ways. Quadratic polynomials curve in a single arc, forming a parabola.

Polynomial functions are versatile mathematical tools essential for analyzing and understanding various types of motions and trends.