Question

If \( A^4 + B^4 = 100 \), then the greatest possible value of A is between

• A. 0 and 3
• B. 3 and 6
• C. 6 and 9
• D. 9 and 12
• E. 12 and 15

Hint:

When solving equations with multiple variables, consider boundary cases (such as setting other variables to zero) to find the maximum or minimum value for the desired variable.

Answer 1

The Correct Option is C

Step 1: Solve the equation for A.
We are given the equation \( A^4 + B^4 = 100 \). To maximize the value of A, we need to consider the scenario when B is minimized. Since \( B^4 \geq 0 \), the minimum value for \( B^4 \) is 0 (when \( B = 0 \)).
Step 2: Set B = 0.
Substituting \( B = 0 \) into the equation, we get: \[ A^4 + 0^4 = 100 \] \[ A^4 = 100 \] Taking the fourth root of both sides: \[ A = \sqrt[4]{100} = \sqrt{10} \approx 3.16 \] Thus, the greatest value for A when \( B = 0 \) is approximately 3.16. Step 3: Conclusion.
Since the greatest value of \( A \) when \( B = 0 \) is about 3.16, we can conclude that the greatest possible value of A is between 3 and 6. Therefore, the correct answer is (B) 3 and 6.