Question

1. a right triangular prism has triangle bases with a base of 6 cm, height of 4 cm, and prism length of 8 cm. find the total surface area. assume its a right triangle. (2 marks) triangular p=6×4×8=198 part c - thinking (2 marks) 5. a cereal box is shaped like a rectangular prism. you want to reduce the amount of cardboard used without changing the volume. explain one way you could reduce the surface area of the box and why it works. (2 marks - explanation and reasoning) part d - communication (2 marks) 6. a student found the surface area of a triangular prism by only calculating the area of one triangle base and one rectangle. explain clearly what the student missed and how to fix the mistake. (2 marks - clarity, detail, correct vocabulary)

Explanation:

Step1: Calculate the area of the triangular bases

The formula for the area of a triangle is \(A=\frac{1}{2}bh\). For the triangular base with \(b = 6\mathrm{cm}\) and \(h=4\mathrm{cm}\), we have \(A_{base}=\frac{1}{2}\times6\times4=12\mathrm{cm}^2\). Since there are 2 triangular bases, \(A_{total - bases}=2\times12 = 24\mathrm{cm}^2\).

Step2: Calculate the length of the hypotenuse of the triangular base

Using the Pythagorean theorem \(c=\sqrt{a^{2}+b^{2}}\), where \(a = 4\mathrm{cm}\) and \(b=6\mathrm{cm}\), \(c=\sqrt{4^{2}+6^{2}}=\sqrt{16 + 36}=\sqrt{52}=2\sqrt{13}\mathrm{cm}\approx 7.21\mathrm{cm}\).

Step3: Calculate the area of the rectangular faces

The three rectangular faces have dimensions:

• One with \(6\times8\), \(A_1=6\times8 = 48\mathrm{cm}^2\)
• One with \(4\times8\), \(A_2=4\times8=32\mathrm{cm}^2\)
• One with \(c\times8\approx7.21\times8 = 57.68\mathrm{cm}^2\)

The total area of the rectangular faces \(A_{rectangles}=48 + 32+57.68=137.68\mathrm{cm}^2\)

Step4: Calculate the total surface area

\(A_{total}=A_{total - bases}+A_{rectangles}=24+137.68 = 161.68\mathrm{cm}^2\)

Answer:

The total surface area of the right - angled triangular prism is approximately \(161.68\mathrm{cm}^2\)