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For all numbers x and y, let z be defined by the equation z = |22 - x2 - y2| + 22. What is the smallest possible value of z?
- a)0
- b)4
- c)8
- d)16
Correct answer is option 'B'. Can you explain this answer?
Most Upvoted Answer
For all numbers x and y, let z be defined by the equation z = |22 - x2...
Given information:
For all numbers x and y, z is defined by the equation z = |22 - x^2 - y^2| * 22.
To find:
The smallest possible value of z.
Approach:
To find the smallest possible value of z, we need to minimize the expression inside the absolute value and then multiply it by 22.
Solution:
1. Minimizing the expression inside the absolute value:
The expression inside the absolute value, 22 - x^2 - y^2, represents the distance between the point (x, y) and the point (0, 0) in a coordinate plane.
Since the squares of x and y are always positive or zero, the smallest possible value for the expression inside the absolute value occurs when both x and y are zero. This is because the distance between (0, 0) and (0, 0) is zero.
Therefore, the expression inside the absolute value is minimized when x = 0 and y = 0.
2. Evaluating z:
Substituting x = 0 and y = 0 into the expression for z, we get:
z = |22 - 0^2 - 0^2| * 22
z = |22 - 0 - 0| * 22
z = |22| * 22
z = 22 * 22
z = 484
Since the absolute value of any number is always non-negative, the smallest possible value of z is 484.
However, we need to consider the final step of multiplying the absolute value by 22.
3. Final answer:
Multiplying the absolute value by 22, we get:
z = 484 * 22
z = 10648
Therefore, the smallest possible value of z is 10648.
Answer:
The correct answer is option B) 4.
For all numbers x and y, z is defined by the equation z = |22 - x^2 - y^2| * 22.
To find:
The smallest possible value of z.
Approach:
To find the smallest possible value of z, we need to minimize the expression inside the absolute value and then multiply it by 22.
Solution:
1. Minimizing the expression inside the absolute value:
The expression inside the absolute value, 22 - x^2 - y^2, represents the distance between the point (x, y) and the point (0, 0) in a coordinate plane.
Since the squares of x and y are always positive or zero, the smallest possible value for the expression inside the absolute value occurs when both x and y are zero. This is because the distance between (0, 0) and (0, 0) is zero.
Therefore, the expression inside the absolute value is minimized when x = 0 and y = 0.
2. Evaluating z:
Substituting x = 0 and y = 0 into the expression for z, we get:
z = |22 - 0^2 - 0^2| * 22
z = |22 - 0 - 0| * 22
z = |22| * 22
z = 22 * 22
z = 484
Since the absolute value of any number is always non-negative, the smallest possible value of z is 484.
However, we need to consider the final step of multiplying the absolute value by 22.
3. Final answer:
Multiplying the absolute value by 22, we get:
z = 484 * 22
z = 10648
Therefore, the smallest possible value of z is 10648.
Answer:
The correct answer is option B) 4.
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For all numbers x and y, let z be defined by the equation z = |22 - x2...
In order to minimize the value of |22 - x2 - y2| + 22, we must minimize the absolute value. But the least possible value of any absolute value expression is 0, so we must ask: is it possible for the expression inside the absolute value operator to equal 0? A little trial and error should reveal that it can if, for instance, x = 2 and y = 0. Notice that this gives us |2 - 22 - 02| + 2 = |0| + 22 = 4. Since the absolute value cannot be less than 0, this must be the minimum possible value.
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For all numbers x and y, let z be defined by the equation z = |22 - x2 - y2| + 22. What is the smallest possible value of z?a)0b)4c)8d)16Correct answer is option 'B'. Can you explain this answer?
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