📌 Step 1: Identify the trig ratio
We know the hypotenuse (200 ft) and want the opposite side (height).
Use sine: sin = opposite / hypotenuse

📌 Step 2: Set up and solve
sin(55°) = h / 200
0.819 = h / 200
h = 200 × 0.819 = 163.8

📌 Answer: ≈ 164 feet

💡 Tip: Angle of elevation from ground = angle between the string and the horizontal, NOT the vertical.

By the displacement principle, a fully submerged solid pushes up a layer of water whose volume equals the solid's volume. That layer is a rectangular prism: V = 24 · 18 · 4 = 1728 cm³. Since the solid is a cube, s³ = 1728, so s = ∛1728 = 12 cm. A cube has 6 congruent square faces, so SA = 6s² = 6 · 144 = 864 cm². The most tempting wrong answer is 720 cm², which comes from counting only 5 faces because the cube 'sits on the bottom' — surface area counts every face of the solid regardless of what it rests against. Answering 144 cm² stops at a single face, and 72 cm² comes from using 6s instead of 6s².

The three vertices adjacent to one corner are not joined by edges of the cube — each pair lies on a common face and is joined by a face diagonal. By the Pythagorean Theorem the face diagonal is √(12² + 12²) = 12√2 cm, so the cross section is an equilateral triangle with all three sides 12√2. Using A = (√3/4)s²: A = (√3/4)(12√2)² = (√3/4)(288) = 72√3 ≈ 124.7 cm². The tempting wrong value 36√3 comes from putting the edge length 12 into (√3/4)s² — that treats the three cut vertices as if they were one edge apart. The value 72 comes from imagining a right triangle with two legs of 12 and taking ½·12·12, which is the area of half a face, not the slanted cross section. The value 144 is the area of a whole face of the cube.

📌 Step 1: Understand permutations
Arranging n objects in a line uses n! (n factorial).

📌 Step 2: Calculate 5!
5! = 5 × 4 × 3 × 2 × 1

📌 Step 3: Solve step by step
5 × 4 = 20
20 × 3 = 60
60 × 2 = 120
120 × 1 = 120

💡 Key difference:
• Permutation = order matters (arranging in a line)
• Combination = order doesn't matter (choosing a team)
• Formula: P(n,r) = n!/(n-r)! and C(n,r) = n!/(r!(n-r)!)

📌 Step 1: Recall the central angle-arc relationship
A central angle equals its intercepted arc.

📌 Step 2: Apply
Central angle = 90° → Arc = 90°

💡 Key relationships in circles:
• Central angle = intercepted arc
• Inscribed angle = ½ × intercepted arc
• Two inscribed angles intercepting the same arc are equal

📌 Step 1: Recall the area formula for a regular polygon
A = ½ × perimeter × apothem

📌 Step 2: Find the perimeter
Perimeter = 6 sides × 6 cm = 36 cm

📌 Step 3: Calculate the area
A = ½ × 36 × 5.2 = 18 × 5.2 = 93.6 cm²

💡 What's an apothem? The distance from the center to the midpoint of a side. It's always perpendicular to the side.

Step 1: Translate 6 right → add 6 to x: W(−4,2) → (−4+6, 2) = (2, 2).
Step 2: Reflect over x-axis → negate y: (2, 2) → (2, −2).
So W′ = (2, −2).

Area of square = 20² = 400 cm².
Area of circle = πr² = 3.14 × 8² = 3.14 × 64 = 200.96 cm².
P = circle/square = 200.96/400 ≈ 0.5024 ≈ 50.2%.

📌 Step 1: Name the rule
Volume of a rectangular prism is V = length × width × height. Each dimension appears exactly once, so scaling one dimension by a factor k scales the volume by that same factor k.

📌 Step 2: Use this question's numbers
Original: V = 4 × 6 × 10 = 240
Only the height changes, 10 → 5:
New: V = 4 × 6 × 5 = 120

📌 Step 3: Compare
120/240 = 1/2, so the new volume is exactly half the original — 240 cubic units becomes 120 cubic units.

📌 Step 4: Check the other fractions both ways
The remaining options name the fractions 1/3, 1/4, and 1/8. Whether you read those as "the volume shrinks to that fraction" or as "the volume shrinks by that much of itself," none of them produce 120. Reading them as shrinks-to gives 80, 60, and 30. Reading them as shrinks-by gives 160, 180, and 210. The computed volume is 120, which matches only the halving statement.

💡 The tempting error: reaching for 1/8. That factor belongs to a different scenario — halving *all three* dimensions, where 2 × 3 × 5 = 30, and (1/2)(1/2)(1/2) = 1/8 because each halved dimension contributes its own factor. This stem changes one dimension only, so there is exactly one factor of 1/2. Count how many dimensions actually changed before you scale: one changed dimension → factor k, two → k², all three → k³. And when a choice describes a change as a fraction, compute the resulting volume yourself and compare numbers rather than matching the fraction by sight.