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a :b for the greatest possible value...

a :b for the greatest possible value of 5 - ( 3a - b)`""^2` is

A

`3 :5`

B

`1 :5`

C

`1 :3`

D

`3 :1`

Text Solution

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The correct Answer is:
To solve the problem, we need to find the ratio \( a : b \) for the greatest possible value of the expression \( 5 - (3a - b)^2 \). ### Step-by-step Solution: 1. **Understanding the Expression**: We start with the expression \( 5 - (3a - b)^2 \). To maximize this expression, we need to minimize the term \( (3a - b)^2 \). 2. **Minimizing the Square**: The minimum value of any square term \( x^2 \) is 0. Therefore, we want to set: \[ (3a - b)^2 = 0 \] This implies that: \[ 3a - b = 0 \] 3. **Solving for b**: Rearranging the equation \( 3a - b = 0 \) gives us: \[ b = 3a \] 4. **Finding the Ratio \( a : b \)**: Now, we can express the ratio \( a : b \): \[ a : b = a : 3a \] This simplifies to: \[ a : b = 1 : 3 \] 5. **Conclusion**: Therefore, the greatest possible value of \( 5 - (3a - b)^2 \) occurs when \( a : b = 1 : 3 \). ### Final Answer: The ratio \( a : b \) for the greatest possible value of \( 5 - (3a - b)^2 \) is \( 1 : 3 \).
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