Grade 8 · Bracket a principal square root · 649 of 750
Bracket a principal square root · practice 76
Enclose √33 after 4 bisections starting from [0, 33]. Report bounds rounded outward to five decimal places and the exact interval width. Follow the calculation, then test the quantities in the animated example.
Try the question, use a hint when you need one, and compare your reasoning with the worked solution. The animation starts with this chapter’s values; changing its controls explores a new case.
01 · Make a prediction
Your practice question
Enclose √33 after 4 bisections starting from [0, 33]. Report bounds rounded outward to five decimal places and the exact interval width.
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A starting point
At each midpoint, compare its square with the radicand and retain the half containing the root.
Work through the reasoning
Step 1
Translate the givens
Start with 0² ≤ 33 ≤ 33². The first midpoint is 16.5, whose square is 272.25.
Step 2
Calculate with the model
Since 272.25 > 33, retain [0, 16.5]. Repeat the same comparison for 4 total bisections.
Step 3
Check and interpret
The final unrounded interval is [4.125, 6.1875]. Each cut halves its width, giving 33/2^4 = 2.0625. Round the lower endpoint down and upper up to obtain [4.125, 6.1875], preserving enclosure.
The answer
Bounds [4.125, 6.1875]; exact width = 33/2^4 = 2.0625.
Compare the method as well as the result. A different valid method may reach the same answer. Keep exact values until the last step when the question asks for rounding.
02 · Explore the model
See the mathematical relationship move.
Use Play, Pause, and the timeline to inspect the construction. Reset restores the question’s original settings. The displayed assumptions describe where this model applies.
Watch the relationship
PausedHD animation studio
From experiment to screen.
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The mathematical idea
A positive square root can be trapped between numbers whose squares lie below and above the target. Bisection halves this interval repeatedly. A short decimal or narrow interval is an approximation; it does not prove that a non-square target has a rational square root. Starting quantities: Radicand n = 33; Number of bisections = 4.
L² ≤ n ≤ U²; midpoint M=(L+U)/2
03 · Reflect and transfer
Explain what changes and why.
Why can rounding both bounds to the nearest decimal destroy a guaranteed enclosure?
This is a distinct guided scenario using a reusable mathematical model. Similar-looking diagrams can represent different given values and conclusions; they are not different mathematical theories.