Question

identify the 3 - d shapes with a surface area less than 100 cm².
a. triangular prism (base area = 9 cm², perimeter of base = 9 cm, height = 5 cm)
b. cylinder (radius = 3 cm, height = 8 cm)
c. rectangular prism (length = 10 cm, width = 4 cm, height = 5 cm)
d. sphere (radius = 3 cm)

Explanation:

Step1: Calculate surface area of triangular prism

The formula for the surface area of a triangular prism is \(SA = 2B+Ph\), where \(B\) is the base area, \(P\) is the perimeter of the base, and \(h\) is the height.
Substitute \(B = 9\), \(P = 9\), and \(h = 5\) into the formula:
\(SA=2\times9 + 9\times5=18 + 45=63\ \text{cm}^2\)

Step2: Calculate surface area of cylinder

The formula for the surface area of a cylinder is \(SA = 2\pi r(r + h)\).
Substitute \(r = 3\) and \(h = 8\) into the formula:
\(SA=2\pi\times3\times(3 + 8)=6\pi\times11\approx6\times3.14\times11 = 207.24\ \text{cm}^2\)

Step3: Calculate surface area of rectangular prism

The formula for the surface area of a rectangular prism is \(SA=2(lw+lh+wh)\).
Substitute \(l = 10\), \(w = 4\), and \(h = 5\) into the formula:
\(SA=2\times(10\times4+10\times5 + 4\times5)=2\times(40 + 50+20)=2\times110 = 220\ \text{cm}^2\)

Step4: Calculate surface area of sphere

The formula for the surface area of a sphere is \(SA = 4\pi r^2\).
Substitute \(r = 3\) into the formula:
\(SA=4\pi\times3^2=4\pi\times9\approx4\times3.14\times9=113.04\ \text{cm}^2\)

Answer:

A. Triangular Prism (Base Area = 9 cm², Perimeter of Base = 9 cm, Height = 5 cm)