Question

math 9 unit 1 shape and space
lesson 3 - 4 quiz: surface area of prisms & cylinders
sa of a cylinder formula: sa = 2\pi r^{2}+2\pi rh
all area calculations must include formula, substitution and solution for full marks. show all work and include proper units for your answer.

1. calculate the surface area of this triangular prism. hint: drawing and labeling pictures of the shapes to help you. include formula, substitution and solution. the hypotenuse of the right - triangle is 17 cm, the height of the triangle is 13 cm, the base of the triangle is 8 cm and the length of the prism is 12 cm.
2. calculate the surface area of this cylinder. the diameter of the circular base is 4 m and the height of the cylinder is 15 m.

Explanation:

Step1: Identify the formula for surface - area of a triangular prism

The surface - area of a triangular prism $SA = 2B+Ph$, where $B$ is the area of the base triangle, $P$ is the perimeter of the base triangle, and $h$ is the height of the prism.
For the base triangle with sides $a = 5$ cm, $b = 8$ cm, and $c = \sqrt{5^{2}+8^{2}}=\sqrt{25 + 64}=\sqrt{89}\approx 9.43$ cm. The area of the base triangle $B=\frac{1}{2}\times5\times8 = 20$ $cm^{2}$, and the perimeter of the base triangle $P=5 + 8+ \sqrt{89}\approx5 + 8+9.43 = 22.43$ cm. The height of the prism $h = 12$ cm.
$SA=2\times20+22.43\times12$
$SA = 40+269.16$
$SA=309.16$ $cm^{2}$

Step2: Identify the formula for surface - area of a cylinder

The surface - area of a cylinder $SA = 2\pi r^{2}+2\pi rh$. Given that the diameter $d = 4$ m, so the radius $r=\frac{d}{2}=2$ m and the height $h = 15$ m.
$SA=2\pi(2)^{2}+2\pi(2)(15)$
$SA = 8\pi+60\pi$
$SA=68\pi$
$SA\approx68\times3.14 = 213.52$ $m^{2}$

Answer:

1. The surface - area of the triangular prism is approximately $309.16$ $cm^{2}$.
2. The surface - area of the cylinder is approximately $213.52$ $m^{2}$.