SLAI Unit 1 Review and Practice Problems

Here is a step-by-step breakdown on how to solve the main types of problems covered in your Unit 1 notes:

1. Measurement Bounds and Percent Error

  • Upper and Lower Bounds:

    1. Identify the unit of precision (e.g., rounded to the nearest 1 cm1\,\text{cm} means precision unit = 1 cm1\,\text{cm}).

    2. Calculate the margin of error: Margin of Error=Precision Unit2\text{Margin of Error} = \frac{\text{Precision Unit}}{2}.

    3. Lower Bound = Measured Value−Margin of Error\text{Measured Value} - \text{Margin of Error}.

    4. Upper Bound = Measured Value+Margin of Error\text{Measured Value} + \text{Margin of Error}.

    5. Express as an inequality: Lower Bound≤True Value<Upper Bound\text{Lower Bound} \le \text{True Value} < \text{Upper Bound}.

  • Percent Error:

    • Use the formula: Percent Error=∣Approximate Value−Exact Value∣Exact Value×100%\text{Percent Error} = \frac{|\text{Approximate Value} - \text{Exact Value}|}{\text{Exact Value}} \times 100\%

2. Trigonometry and Bearings

  • Understanding Bearings:

    • Always measure clockwise starting from True North (000∘000^\circ).

    • Always write bearings with 3 digits (e.g., 036∘036^\circ).

  • Choosing the Right Formula:

    • Right Triangles: Use SOH CAH TOA\text{SOH CAH TOA}.

    • Non-Right Triangles:

    • Law of Cosines: Use c2=a2+b2−2abcos⁡(C)c^2 = a^2 + b^2 - 2ab \cos(C) when you know two sides and the included angle (SAS), or all three sides (SSS).

    • Law of Sines: Use asin⁡(A)=bsin⁡(B)\frac{a}{\sin(A)} = \frac{b}{\sin(B)} when you have a matching side-angle pair.

    • Triangle Area: Use Area=12absin⁡(C)\text{Area} = \frac{1}{2} ab \sin(C).

3. Circle Geometry

  • Arc Length:

    • Arc Length=θ360∘×2πr\text{Arc Length} = \frac{\theta}{360^\circ} \times 2 \pi r

  • Sector Area:

    • Areasector=θ360∘×πr2\text{Area}_{\text{sector}} = \frac{\theta}{360^\circ} \times \pi r^2

  • Segment Area (Shaded Region):

    • Areasegment=Areasector−Areatriangle=(θ360∘×πr2)−(12r2sin⁡(θ))\text{Area}_{\text{segment}} = \text{Area}_{\text{sector}} - \text{Area}_{\text{triangle}} = \left( \frac{\theta}{360^\circ} \times \pi r^2 \right) - \left( \frac{1}{2} r^2 \sin(\theta) \right)