A packet of sweets is in the shape of a closed triangular-based prism - Junior Cycle Mathematics - Question 12 - 2016

To find the side marked x cm, we will use trigonometric functions:

1. Identify the triangle parameters:

• Given angles 70°, 70° and the adjacent side of 7 cm to the angle.

2. Apply the cosine function:

• From the cosine definition, we have: $ext{cos}(70°) = \frac{\text{adjacent}}{\text{hypotenuse}}$
• Hence, we calculate: $x = \frac{7}{\text{cos}(70°)}$

3. Compute the value:

• Performing the calculation yields: $x ≈ 7.46 ext{ cm (to two decimal places)}$

To find the areas of the faces A, B, and C:

1. Calculate area A:

• Face A is a rectangle with dimensions 7 cm (base) and 12 cm (height).
• Area of A:
  $A = 7 \times 12 = 84 \text{ cm}²$

2. Calculate area B:

• Face B is also a rectangle with dimensions of 12 cm and the length found for side x:
• Area of B: $B = 12 \times x = 12 \times 10.23 = 123 \text{ cm}² ext{ (to nearest cm²)}$

3. Calculate area C:

• Face C requires determining the perpendicular height for its triangle form.
• Using layers of trigonometry or calculating the base and height gives: $C = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 7 \times 9.616 ≈ 34 \text{ cm}²$