Question

1. a) find the surface area of each prism to the nearest tenth of a square centimetre. (image of a rectangular prism with dimensions 18.5 cm, 13.5 cm, 13.5 cm)

Explanation:

Step1: Calculate the area of the two congruent triangular bases

The base of the triangular base \(b = 13.5\) cm. Let's assume the triangular base is a right - angled triangle (since it's a prism with right - angled side - like structure in terms of given dimensions). Using the Pythagorean theorem for the triangular base: if the two legs of the right - triangle are \(a = 13.5\) cm and \(b = 13.5\) cm, then the hypotenuse \(c=\sqrt{13.5^{2}+13.5^{2}}=\sqrt{2\times13.5^{2}} = 13.5\sqrt{2}\approx19.09\) cm. But for the area of the triangular base \(A_{base}=\frac{1}{2}\times13.5\times13.5=\frac{1}{2}\times182.25 = 91.125\) \(cm^{2}\). The area of the two triangular bases \(A_{bases}=2\times91.125 = 182.25\) \(cm^{2}\).

Step2: Calculate the area of the three rectangular lateral faces

The three rectangular lateral faces:

• One with dimensions \(13.5\times18.5\): \(A_{1}=13.5\times18.5 = 249.75\) \(cm^{2}\)
• Another with dimensions \(13.5\times18.5\): \(A_{2}=13.5\times18.5 = 249.75\) \(cm^{2}\)
• The third with dimensions \(13.5\sqrt{2}\times18.5\approx19.09\times18.5 = 353.165\) \(cm^{2}\)

The sum of the lateral face areas \(A_{lateral}=249.75 + 249.75+353.165=852.665\) \(cm^{2}\)

Step3: Calculate the total surface area

The total surface area \(A = A_{bases}+A_{lateral}\)
\(A=182.25 + 852.665=1034.915\approx1034.9\) \(cm^{2}\)

Answer:

\(1034.9\) \(cm^{2}\)