Problem 52 - Algebra A Combined Function

Question

The Mars Odyssey spacecraft was launched in 2001 , beginning a multiyear mission to observe and map the planet Mars. Mars Odyssey was launched on Boeing's Delta II 7925 launch vehicle using nine strap-on solid rocket motors. Each solid rocket motor has a height that is 8 meters more than 5 times its diameter. If the sum of the height and the diameter for a single solid rocket motor is 14 meters, find each dimension. (Recently, NASA approved a continuation of the Odyssey mission through September 2010.) (Source: NASA)

Step-by-Step Solution

Answer

The diameter is 1 meter, and the height is 13 meters.

Step 1: Define Variables

Let the diameter of the solid rocket motor be \( d \) meters and the height be \( h \) meters.

Step 2: Express Height in Terms of Diameter

According to the problem, the height is 8 meters more than 5 times the diameter. Express this as an equation: \( h = 5d + 8 \).

Step 3: Express Relationship between Height and Diameter

The sum of the height and diameter is 14 meters. Thus, we have the equation: \( h + d = 14 \).

Step 4: Substitute and Solve for Diameter

Substitute the expression for height from Step 2 into the equation from Step 3: \( (5d + 8) + d = 14 \). Simplify and solve for \( d \): \( 6d + 8 = 14 \). Subtract 8 from both sides: \( 6d = 6 \). Divide both sides by 6: \( d = 1 \).

Step 5: Find the Height

Use the value of \( d \) to find \( h \) using the equation from Step 2: \( h = 5(1) + 8 = 5 + 8 = 13 \).

Step 6: Verify the Solution

Check that the values of \( d = 1 \) and \( h = 13 \) satisfy both original conditions. The sum \( h + d = 13 + 1 = 14 \), and the equation for height is valid: \( h = 5 \times 1 + 8 = 13 \). Therefore, the solution is correct.

Key Concepts

Equation Solving

Equation solving is a fundamental skill in algebra that involves finding the values of variables that satisfy given equations. In this exercise, two equations are set up based on the descriptions of the height and diameter of a solid rocket motor.
To solve equations:

• Identify the variables in the problem.
• Write equations that represent the relationships between these variables.
• Use algebraic techniques to isolate the variables and solve for their values.

By following these steps, you can systematically approach and solve algebraic problems effectively.
Breaking down the problem into manageable steps allows you to focus on one part of the equation at a time, simplifying the process of finding a solution.

Linear Equations

Linear equations are equations where the highest power of the variable is one. They appear as straight lines when graphed on a coordinate plane.
In this exercise, we encountered linear equations in the form of:

• Height equation: \( h = 5d + 8 \)
• Sum equation: \( h + d = 14 \)

These equations involve the variables height \( h \) and diameter \( d \) and are examples of linear relationships.
The key to solving these linear equations is understanding how to transform, manipulate, and use them to find the value of the variable of interest.
Once you get comfortable with linear equations, solving more complex problems becomes easier because many mathematical problems can be simplified into linear forms.

Variable Definition

Defining variables is a crucial first step in solving algebra problems. This process involves deciding which symbols to use to represent the quantities in the problem.
In the given exercise:

• The diameter of the solid rocket motor is represented by \( d \).
• The height is represented by \( h \).

Choosing intuitive symbols can help in keeping your work organized and reducing confusion throughout the solving process.
Once variables are defined, they serve as the placeholders in equations, helping to express complex relationships and solve for unknown quantities. Solid definitions provide clarity and direction for constructing the equations necessary to find the solution.

Substitution Method

The substitution method is a technique used to solve a system of equations. It involves expressing one variable in terms of another and then substituting this expression into another equation.
In this exercise, we used the substitution method:

• First, express the height \( h \) in terms of diameter \( d \) using the equation \( h = 5d + 8 \).
• Then, substitute this expression into the sum equation \( h + d = 14 \).
• This simplifies the problem to one equation in one variable, making it easier to solve for \( d \).

Finally, use the found value of \( d \) to find \( h \) by substituting back into the height equation.
The substitution method is powerful because it systematically reduces the number of unknowns, allowing you to solve complex systems step by step.