Math Problem Statement
The sediment carrying capacity of the stream (S) varies with its velocity (v) raised to the 6th power. When the stream velocity =1, the carrying capacity equals 0.1, and when the velocity = 2, the carrying capacity equals 6.4. Which of the following functions represents S(v)? Select one: S(v)=0.1v6 S(v)=|v+6| S(v)=v16−0.9 S(v)=6v+0.
Solution
To determine which function represents the sediment carrying capacity , we need to identify a relationship where the sediment carrying capacity varies with the stream velocity raised to the 6th power. We are given the following information:
- When the velocity , the carrying capacity .
- When the velocity , the carrying capacity .
The problem suggests that varies with raised to the 6th power. This implies the general form:
where is a constant that we need to determine using the given data.
Step 1: Use the given data points
We know:
- When , . Plugging these values into the equation :
So the equation becomes:
Step 2: Verify with the second data point
We are also told that when , . Let's verify if this equation holds true for this second data point:
Since this is correct, the equation is verified.
Conclusion:
The function that represents is:
Therefore, the correct answer is:
S(v) = 0.1v^6
Would you like further details on the derivation or clarification on another part of the problem? Here are some related questions you could explore:
- How does the exponent in the equation affect the relationship between velocity and sediment carrying capacity?
- If the exponent were different, how would that change the carrying capacity as the velocity changes?
- How would the equation change if the constant were different?
- What would happen to the carrying capacity if the velocity approached zero?
- How would this model behave for negative velocities?
Tip: Always verify the formula with multiple data points to ensure the model fits the situation correctly.
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Math Problem Analysis
Mathematical Concepts
Algebra
Exponents
Functions
Formulas
S(v) = k * v^6
Theorems
Power functions
Suitable Grade Level
Grades 9-12
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