Question

The mass of the particles that a river can transport is proportional to the sixth power of the speed of the river. A certain river normally flows at a speed of 1 meter per second. What must its speed be in order to transport particles that are twice as massive as usual? 10 times as massive? 100 times as massive?

Short answer

The speeds required to carry particles that are twice, ten times, and hundred times as massive as usual are approximately 1.12 m/s, 1.47 m/s, and 2.15 m/s respectively.

Step-by-step solution

Given Information

The given information is: The mass of particles 'm' that the river can transport is proportional to the 6th power of the speed 'v' of the river, this is denoted as 'm' \( \propto \) 'v'^6 or 'm' = k 'v'^6. A certain river normally flows at a speed of 1 meter per second (Normal Speed). We are asked to compute the speeds required to transport particles twice, ten times, and hundred times as massive as usual.

Compute the Proportionality Constant 'k'

From the normal condition of the river, mass m1 is transported by speed v1 = 1 m/s. From this, the proportionality constant 'k' can be calculated using the formula 'k' = 'm1' / 'v1'^6. As no actual value for m1 is given, consider m1 to be 'm' for convenience.

Compute Required Speeds for Increased Masses

Now, to find the speeds v2, v10, v100 necessary to transport masses that are twice (2m), ten times (10m), and hundred times (100m) as massive as 'm', we can use the formula v = (m/k)^(1/6). Given that k = m, the equations will simplify to v2 = (2m/m)^(1/6), v10 = (10m/m)^(1/6) and v100 = (100m/m)^(1/6). Calculating these will give the respective speeds.

Simplifying Calculations

Considering that (m/m) simplifies to 1, the speed calculations become: v2 = 2^(1/6), v10 = 10^(1/6) and v100 = 100^(1/6). These are the speeds of the river needed to transport particles that are twice, ten times, and hundred times as massive as usual.

Key concepts

Proportional Reasoning

Understanding proportional reasoning is fundamental when it comes to solving real-world problems like predicting the mass of particles that a river can transport. Imagine a see-saw, with weights balancing on either ends. If one side is altered, the other side must change to maintain balance. Proportional reasoning is similar; it involves an understanding that two quantities vary in a way that if one quantity changes, the other does too, maintaining a constant ratio.

Let's apply this idea to our river problem. The mass of the particles is directly related to the speed of the river at an exponent of six. If the river's speed increases, the mass of particles it can transport also increases, and it does so in a very specific way, following the formula mentioned in our solution steps. The key takeaway is that by understanding the constant ratio between mass and speed, we can predict the needed speed for different mass increases with precision.

Here's a quick tip: In problems like these, look for keywords such as 'proportional to', 'directly proportional', or 'varies as' to identify proportional relationships.

Exponentiation in Algebra

The concept of exponentiation in algebra arises frequently, and it's integral to understand this operation when dealing with problems involving growth, decay, or any non-linear scale-up. Exponents signify repeated multiplication, and when you see a term like 'v'^6, it means 'v' multiplied by itself six times.

In the river problem, the speed of the river to an exponent of six determines the mass of particles transported—a clear example where exponentiation helps us model situations that wouldn't be as easily represented with just multiplication. It's helpful to remember that exponentiation is a shortcut for expressing large numbers or small fractions, and it dramatically affects calculations. When the speed doubles, the mass doesn't just double; it increases by a factor of 2^6, which is 64 times greater!

Breaking Down Large Exponents

For large exponents, such as determining the speed to transport 100 times the mass, it's beneficial to understand the laws of exponents. The expression (100m/m)^(1/6) simplifies because any number divided by itself is 1, leaving us with 100^(1/6). This is a perfect cube of (10^(1/6))^2, which could simplify calculation.

Mathematical Problem Solving

The art of mathematical problem solving isn't just about crunching numbers; it's a process that involves understanding the problem, devising a plan, carrying out the plan, and then looking back. This approach was illuminated by our steps for solving the river speed and mass exercise.

Understanding comes first. In our case, it's about grasping the proportional relationship and the effect of exponentiation. Next, we devise a plan by setting up equations based on the understood principles. Carrying out the plan involves calculations, which should be approached systematically to avoid mistakes. Finally, looking back is an excellent practice to check your work. Could there be a simpler method? Have you considered all factors?

Improving in mathematical problem solving involves practice and a willingness to tackle problems from multiple angles. Remember, the goal is not just to find the answer, but to understand the 'why' and 'how' behind it, ensuring that the learned methods can be applied to novel problems you might encounter in the future.