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 S(v)S(v), 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 v=1v = 1, the carrying capacity S=0.1S = 0.1.
  • When the velocity v=2v = 2, the carrying capacity S=6.4S = 6.4.

The problem suggests that S(v)S(v) varies with vv raised to the 6th power. This implies the general form:

S(v)=kv6S(v) = k \cdot v^6 where kk is a constant that we need to determine using the given data.

Step 1: Use the given data points

We know:

  • When v=1v = 1, S=0.1S = 0.1. Plugging these values into the equation S(v)=kv6S(v) = k \cdot v^6:

0.1=k16    k=0.10.1 = k \cdot 1^6 \implies k = 0.1

So the equation becomes:

S(v)=0.1v6S(v) = 0.1 \cdot v^6

Step 2: Verify with the second data point

We are also told that when v=2v = 2, S=6.4S = 6.4. Let's verify if this equation holds true for this second data point:

S(2)=0.126=0.164=6.4S(2) = 0.1 \cdot 2^6 = 0.1 \cdot 64 = 6.4

Since this is correct, the equation is verified.

Conclusion:

The function that represents S(v)S(v) is:

S(v)=0.1v6S(v) = 0.1 \cdot v^6

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:

  1. How does the exponent in the equation affect the relationship between velocity and sediment carrying capacity?
  2. If the exponent were different, how would that change the carrying capacity as the velocity changes?
  3. How would the equation change if the constant kk were different?
  4. What would happen to the carrying capacity if the velocity approached zero?
  5. 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