Math  /  Numbers & Operations

QuestionWhich sign makes the statement true? π2?8-\frac{\pi}{2} ?-\sqrt{8}

Studdy Solution

STEP 1

1. The expression involves comparing the values of π2-\frac{\pi}{2} and 8-\sqrt{8}.
2. Since both terms are negative, we need to determine which negative number is larger in magnitude (i.e., closer to zero).
3. π\pi is a known constant approximately equal to 3.141593.14159.
4. 8\sqrt{8} is a known value that can be simplified to 222\sqrt{2}, and 2\sqrt{2} is approximately 1.4141.414.

STEP 2

1. Calculate the approximate value of π2-\frac{\pi}{2}.
2. Calculate the approximate value of 8-\sqrt{8}.
3. Compare the two values to determine which is larger in magnitude.
4. Determine the appropriate inequality sign to make the statement true.

STEP 3

Calculate the value of π2\frac{\pi}{2} using the approximation π3.14159\pi \approx 3.14159.
π23.1415921.5708 \frac{\pi}{2} \approx \frac{3.14159}{2} \approx 1.5708

STEP 4

Determine the value of π2-\frac{\pi}{2}.
π21.5708 -\frac{\pi}{2} \approx -1.5708

STEP 5

Simplify and calculate the value of 8\sqrt{8}.
8=22and21.414 \sqrt{8} = 2\sqrt{2} \quad \text{and} \quad \sqrt{2} \approx 1.414 222×1.4142.828 2\sqrt{2} \approx 2 \times 1.414 \approx 2.828

STEP 6

Determine the value of 8-\sqrt{8}.
82.828 -\sqrt{8} \approx -2.828

STEP 7

Compare π21.5708-\frac{\pi}{2} \approx -1.5708 and 82.828-\sqrt{8} \approx -2.828.
Since 1.5708-1.5708 is closer to zero than 2.828-2.828, π2-\frac{\pi}{2} is greater than 8-\sqrt{8}.

STEP 8

The appropriate inequality sign to make the statement true is >>.
π2>8 -\frac{\pi}{2} > -\sqrt{8}
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