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Answer :
To answer the question, we'll begin by calculating the circumference of a circle in different scenarios and then determine the perimeter of a right triangle.
1. Calculate the circumference of a circle:
The formula for the circumference [tex]C[/tex] of a circle is:
[tex]C = \pi \, \text{d}[/tex]
or
[tex]C = 2\pi \, \text{r}[/tex]
where [tex]d[/tex] is the diameter and [tex]r[/tex] is the radius of the circle.
(a) Diameter = 42 cm:
[tex]C = \pi \times 42 = 42\pi[/tex] cm
Approximating [tex]\pi[/tex] as 3.14, [tex]C \approx 42 \times 3.14 = 131.88[/tex] cm
(b) Radius = 12 cm:
[tex]C = 2 \times \pi \times 12 = 24\pi[/tex] cm
Approximating [tex]\pi[/tex] as 3.14, [tex]C \approx 24 \times 3.14 = 75.36[/tex] cm
(c) Diameter = 2.9 km:
[tex]C = \pi \times 2.9 = 2.9\pi[/tex] km
Approximating [tex]\pi[/tex] as 3.14, [tex]C \approx 2.9 \times 3.14 = 9.106[/tex] km
(d) Radius = 72.5 mm:
Convert radius from mm to cm for consistency:
72.5 mm = 7.25 cm
[tex]C = 2 \times \pi \times 7.25 = 14.5\pi[/tex] cm
Approximating [tex]\pi[/tex] as 3.14, [tex]C \approx 14.5 \times 3.14 = 45.53[/tex] cm
2. Calculate the perimeter of the triangle:
For a right triangle, we use the Pythagorean theorem to find the hypotenuse [tex]c[/tex], and then sum all sides to get the perimeter.
(a) Right triangle with sides 18 cm and 3 m:
First, convert 3 m to cm: 3 m = 300 cm
Using the Pythagorean theorem, [tex]c = \sqrt{(18^2 + 300^2)}[/tex]
[tex]c = \sqrt{324 + 90000} = \sqrt{90324} \approx 300.54 \text{ cm}[/tex]
Perimeter [tex]P = 18 + 300 + 300.54 = 618.54[/tex] cm
(b) Right triangle with sides 4 cm and 1.8 cm:
Using the Pythagorean theorem, [tex]c = \sqrt{(4^2 + 1.8^2)}[/tex]
[tex]c = \sqrt{16 + 3.24} = \sqrt{19.24} \approx 4.39 \text{ cm}[/tex]
Perimeter [tex]P = 4 + 1.8 + 4.39 = 10.19[/tex] cm
By following these steps, we can accurately determine both the circumference of the circles and the perimeter of the triangles.
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