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

To find the length of side $x$:

1. Identify the triangle:

   • This is a triangle where the two angles at the base are both $70^ ext{o}$, and the opposite side is 7 cm.

2. Apply the sine rule:

   • The sine of an angle in a right triangle is defined as:

   $ext{sin } heta = \frac{ ext{opposite}}{ ext{hypotenuse}}$

3. Set up the equation:

   • We can express $x$ as:

   $\text{sin } 70^ ext{o} = \frac{7}{x}$

   $x = \frac{7}{\text{sin } 70^ ext{o}}$

4. Calculate the value:

   • Using a calculator:

   $x \approx 10.23 ext{ cm (to two decimal places)}$

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

1. Face A:
   • Dimensions: 12 cm x 7 cm.
   • Area = $12 \times 7 = 84 ext{ cm}^2$.
2. Face B:
   • Dimensions: 12 cm x 7 cm. (Note: Same dimensions as Face A)
   • Area = $12 \times 7 = 84 ext{ cm}^2$.
3. Face C:
   • Here we will use the base (calculated as $\sqrt{3^2 + 7^2}$) as the base length, which is about 7.615 cm.
   • So, the height is 12 cm.
   • Area = $\frac{1}{2} \times \text{base} \times \text{height} \approx \frac{1}{2} \times 7.615 \times 12 \approx 91.38 ext{ cm}^2$.
   • Finally rounding to nearest cm²: 91 cm².