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Chapter 8: Problem 41

Archery. The Olympic flame tower at the 1992 Summer Olympics was lit at a height of about 27 m by a flaming arrow that was launched about 63 m from the base of the tower. If the arrow landed about 63 m beyond the tower, find a quadratic function that expresses the height h of the arrow as a function of the distance d that it traveled horizontally.

Short Answer

Expert verified
The quadratic function is \( h(d) = -\frac{1}{147}d^2 + \frac{6}{7}d \).

Step by step solution

01

Set up the coordinate system

Place the origin of the coordinate system at the base of the tower. Let the horizontal distance d be the x-axis and height h be the y-axis.
02

Determine the form of the quadratic function

The height h as a function of distance d can be described by a quadratic function: \[ h(d) = ad^2 + bd + c \]
03

Identify known points

The arrow is launched from (0, 0), reaches its maximum height at (63, 27), and lands at (126, 0). Thus, you have the points (0, 0), (63, 27), and (126, 0).
04

Use the point (0,0) to determine c

Substituting the point (0, 0) into the quadratic function gives: \[ 0 = a(0)^2 + b(0) + c \] Therefore, c = 0.
05

Set up equations using the other points

Use the points (63, 27) and (126, 0) to form two equations: \[ 27 = a(63)^2 + b(63) \] and \[ 0 = a(126)^2 + b(126) \]
06

Solving for a and b

First, solve the second equation for b: \[ 0 = 15876a + 126b \] \[ b = -126a \]Next, substitute b = -126a into the first equation: \[ 27 = 3969a + 63(-126a) \] \[ 27 = 3969a - 7938a \] \[ 27 = -3969a \] \[ a = -\frac{27}{3969} = -\frac{1}{147} \]Substitute a back into b: \[ b = -126 \cdot -\frac{1}{147} = \frac{126}{147} = \frac{6}{7} \]
07

Write the quadratic function

Now that you have determined a and b, write the quadratic function as: \[ h(d) = -\frac{1}{147}d^2 + \frac{6}{7}d \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding the Coordinate System
To solve this problem, we first need to understand the coordinate system. The coordinate system is a way to describe the position of points by using two numbers, which we call coordinates. In this problem, we use a 2D coordinate system with the horizontal distance (d) along the x-axis and the height (h) along the y-axis. By placing the origin (0,0) at the base of the Olympic flame tower, we can describe where the arrow starts, reaches its maximum height, and lands.
Solving Quadratic Equations
A quadratic equation is a mathematical expression where the highest power of the variable is a square. In this problem, the height of the arrow (h) as a function of the distance (d) it travels is expressed as a quadratic function: \[ h(d) = ad^2 + bd + c \]We have three points: (0,0), (63,27), and (126,0). By using these points, we substitute them into the quadratic function form to find the coefficients a, b, and c. This involves setting up equations based on these points and solving for the unknowns. The calculations step by step are crucial to find the values of a and b, while c is determined directly.
Maximum Height in Projectile Motion
Projectile motion refers to the motion of an object thrown or projected into the air, subject to only the acceleration of gravity. For the arrow, its highest point or maximum height is when it reaches 27 m at a horizontal distance of 63 m. Identifying this peak point is key because it tells us where and when the arrow's vertical speed is zero. At this maximum height, the quadratic function helps us understand how the height changes with the horizontal distance. We use the information to solve for the constants in our quadratic equation, confirming that the maximum height is exactly mid-way horizontally in projectile problems.
Real-world Applications of Algebra
Quadratic equations are not just theoretical; they have numerous real-world applications. Understanding how algebra helps solve real-life problems like the archery example enables us to model situations involving projectile motion, such as sports trajectories, engineering projects, and even safety calculations. In this example, launching an arrow to light an Olympic flame tower shows how accurately modeling with quadratics ensures that the arrow reaches the correct height at the right distance. Mastering these algebraic concepts can lead to practical skills in various fields such as physics, engineering, and even economics where parabolic trends are common.

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