Question

A space shuttle achieves orbit at 9: 23 A.M. At 9: 31 A.M. it has traveled \(2,309.6\) miles in orbit. Find the rate of change in miles per minute.

Short answer

The rate of change, or the speed of the space shuttle, is 288.7 miles per minute.

Step-by-step solution

Identify the given values

The distance that the space shuttle has traveled is 2309.6 miles, and the time it has taken is from 9:23 A.M. to 9:31 A.M. Calculate the time difference in minutes, which is 31 - 23 equals to 8 minutes.

Calculating the rate

The rate or speed is calculated by dividing the total distance by the time taken. That is \(\frac{2309.6 \text{ miles}}{8 \text{ minutes}}\).

Solve the division

Divide 2309.6 by 8 to find the rate.

Key concepts

Linear Functions

Linear functions are a foundational concept in algebra and represent relationships with a constant rate of change. In a mathematical context, these functions are often written in the form of an equation as y = mx + b, where m is the slope, representing the rate of change, and b is the y-intercept.

For instance, when we look at the textbook problem regarding the space shuttle, the distance traveled over time can be represented by a linear function since the shuttle travels at a constant rate in orbit. The slope here represents the rate of speed per minute, and if time is plotted on the x-axis and distance on the y-axis, we would see a straight line, indicating that the motion is uniform and thus can be modeled by a linear function.

Understanding linear functions is crucial in problem solving, as it allows students to predict outcomes and make connections with real-world scenarios, like calculating the distance a space shuttle travels in a given amount of time.

Unit Rate

Unit rate is a comparative value expressing the cost per single unit of measure and it is exceedingly useful for comparison and understanding rates of change in various contexts. In algebra, unit rates can be found by setting a comparison between two different quantities where one of the values is fixed to the unit of one.

In the context of our problem with the space shuttle, we are asked to find out how many miles the shuttle covers in one minute. To compute this, we divide the total distance traveled by the total time taken, effectively finding the unit rate of the shuttle's speed, which is expressed as miles per minute. Understanding and calculating unit rates enable students to translate complex problems into comprehensible real-world interpretations, solidifying their grasp of how to manipulate and apply algebraic equations effectively.

Problem-solving in Algebra

Problem-solving in algebra involves identifying the unknowns, forming equations, and manipulating these equations to find the solutions. To solve algebraic problems, it's essential to break them down into manageable steps and to understand the underlying concepts such as linear functions and rates.

In the space shuttle example, problem-solving begins with identifying the given values (distance and time), continues with forming an equation for rate of change, and concludes with the calculation, which leads to the answer. This structured approach allows one to apply algebra to a variety of real-world situations and to develop critical thinking skills that go beyond rote memorization.

Effective algebraic problem-solving is adaptive, iterative, and logical. It calls for critical analysis of given information, perfecting operations with numeric and algebraic expressions, and the application of concepts to find valid solutions, much like meticulously determining the speed of a space shuttle in orbit.