Discussion Assignments: Scientific Notation Is A Common Way
Discussion Assignmentscientific Notation Is A Common Way Of Writing V
Discussion Assignment: Scientific notation is a common way of writing very large numbers. In everyday life, we see this when dealing with cell phone storage. If a phone has 10 gigabytes of hard drive space, that means it has 10×109 bytes of space. A megabyte is 1×106 bytes.
1. If our phone with 10 gigabytes of free storage downloads a game that takes up 76 megabytes, how much free storage is left?
2. Describe the process of subtracting numbers in scientific notation and give the solution.
3. Discuss other places we frequently see scientific notation in real life?
Part 2 Discussion Assignment
The quadratic equations are common in many types of applications. The quadratic equation gives us a powerful tool to use to solve them. If one throws a ball down from a high cliff, the distance it travels can be modeled by the equation: d(t) = -9.8t2 + 15t + 100 where t is the time in seconds and d is the distance in meters.
1. At what time will the ball hit the ground?
2. You will get two answers because this is a quadratic equation. Do both make sense? (Explain in detail why or why not).
Paper For Above Instruction
The assignment explores the application of scientific notation in everyday life and the mathematical modeling using quadratic equations. Scientific notation serves as an efficient method for representing very large or very small numbers, crucial in fields like technology, science, and engineering. Understanding how to manipulate these numbers, especially through operations like subtraction, is essential for precise calculations in real-world scenarios.
Application of Scientific Notation in Real Life
In digital technology, data storage capacities are expressed using scientific notation due to the enormous numbers involved. For example, a 10-gigabyte storage device equates to 10×109 bytes because 1 gigabyte equals 109 bytes. Similarly, a megabyte, which is 1×106 bytes, helps quantify smaller storage units. When a game occupying 76 megabytes is downloaded onto this device, subtracting the used space from the total storage involves working with these numbers in scientific notation.
The process of subtracting numbers in scientific notation begins with aligning the exponents. For instance, to subtract 76×106 bytes from 10×109 bytes, we rewrite both numbers with the same exponent. Since 109 equals 1×109 and 76×106 equals 0.076×109, the subtraction becomes 10×109 – 0.076×109 = (10 – 0.076)×109 = 9.924×109 bytes. This calculation reveals approximately 9.924 gigabytes of free storage left after the download.
Beyond digital storage, scientific notation is widely used in fields like astronomy, where distances between celestial bodies span vast ranges, or in chemistry for representing concentrations and quantities at atomic scales. The notation simplifies calculations and communication concerning such extreme values.
Modeling Physical Phenomena with Quadratic Equations
Quadratic equations are integral in modeling physical phenomena such as projectile motion. For example, when throwing a ball downward from a high cliff, the distance d(t) it travels over time t can be described by the quadratic function d(t) = -9.8t2 + 15t + 100, where d is measured in meters and t in seconds. The coefficient -9.8 represents acceleration due to gravity, and the initial velocity and height are modeled by the linear and constant terms, respectively.
To determine when the ball hits the ground, the key is to find the time t when d(t) = 0. Solving the quadratic equation -9.8t2 + 15t + 100 = 0 involves applying the quadratic formula: t = [-b ± √(b2 - 4ac)] / 2a, where a = -9.8, b = 15, and c = 100.
Calculating the discriminant: Δ = 152 - 4×(-9.8)×100 = 225 + 3920 = 4145. Using the quadratic formula:
t = [-15 ± √4145] / (2× -9.8)
This yields two solutions: one positive and one negative. Since negative time does not have physical meaning in this context, only the positive root corresponds to the actual time when the ball hits the ground. The positive solution provides a realistic estimate, approximately around 2.07 seconds.
The presence of two solutions is typical in quadratic equations, representing two points in time where the projectile intersects a particular height—here, the ground. Only the positive root makes sense physically, as negative time would imply a moment before the ball was released, which is not possible in the real-world scenario.
Conclusion
Understanding scientific notation and quadratic equations is vital across various disciplines. Scientific notation simplifies handling and communicating extremely large or small numbers, while quadratic equations facilitate precise modeling of physical phenomena. Their applications range from digital storage calculations to predicting the behavior of moving objects, demonstrating their fundamental role in science, technology, and engineering.
References
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- Stewart, J. (2015). Precalculus: Concepts and Models. Cengage Learning.
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- U.S. Department of Energy. (2020). Energy Storage and Data Centers. https://www.energy.gov
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