Consider the equation K  =  (  5R/S  )^N and assume N  =  3. How does K change if S decreases by 10%?

Given the equation \( K = \left( \frac{5R}{S} \right)^N \) and \( N = 3 \), the equation simplifies to:

\[
K = \left( \frac{5R}{S} \right)^3
\]

Now, if \( S \) decreases by 10%, the new value of \( S \) becomes \( S_{\text{new}} = 0.9S \).

Substitute this new \( S_{\text{new}} \) into the equation for \( K \):

\[
K_{\text{new}} = \left( \frac{5R}{0.9S} \right)^3
\]

This can be rewritten as:

\[
K_{\text{new}} = \left( \frac{5R}{S} \times \frac{1}{0.9} \right)^3 = \left( \frac{5R}{S} \right)^3 \times \left( \frac{1}{0.9} \right)^3
\]

We know \( \left( \frac{5R}{S} \right)^3 = K \), so:

\[
K_{\text{new}} = K \times \left( \frac{1}{0.9} \right)^3
\]

Now, calculate \( \left( \frac{1}{0.9} \right)^3 \):

\[
\left( \frac{1}{0.9} \right)^3 = \left( \frac{10}{9} \right)^3 = \frac{1000}{729} \approx 1.372
\]

Thus, the new value of \( K \) is:

\[
K_{\text{new}} \approx 1.372 \times K
\]

Therefore, \( K \) increases by approximately 37.2% when \( S \) decreases by 10%.

Would you like more details on this? Here are 5 related questions to explore:

1. How does \( K \) change if \( S \) increases by 10% instead?
2. What is the general impact on \( K \) if \( S \) changes by a given percentage?
3. How does the value of \( N \) affect the sensitivity of \( K \) to changes in \( S \)?
4. If \( R \) also changes, how would that impact \( K \)?
5. Can you derive the general formula for the percentage change in \( K \) when \( S \) changes by a specific percentage?

**Tip**: When dealing with percentage changes, always consider how the powers of terms influence the overall result!

Consider the equation K = (5R/S)^N and assume N = 3. How does K change if S decreases by 10%?

Explore how K changes when S decreases by 10% in the equation K = (5R/S)^3. Learn about the effects of percentage changes on exponential expressions, with a detailed step-by-step solution and a final answer showing a 37.2% increase in K. Suitable for Grades 10-12 students.

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