contestada

. A rope is swinging in such a way that the length of the arc is

decreasing geometrically. If the first arc is 24 feet long and the third

arc is 6 feet long, what is the length of the second arc?

Respuesta :

ujalakhan18

Answer:

[tex]\huge\boxed{\sf g_2 = 12\ feet}[/tex]

Step-by-step explanation:

The pattern of the arc:

24, g₂, 6, .....

Since the pattern in decreasing geometrically, we will use the formula:

[tex]a_n=ar^{n-1}[/tex]

Where,

n = position of the term

[tex]a_n[/tex] = nth term

a = 1st term

r = common ratio (ratio of second to first term)

Solution:

First, we'll find r.

For 3rd term:

[tex]a_3[/tex] = 6

a = 24

n = 3

r = ?

So,

[tex]\displaystyle a_3=(24)(r)^{3-1}\\\\6 = 24(r^2)\\\\Divide \ both \ sides \ by \ 24\\\\\frac{6}{24} = r^2\\\\\frac{1}{4} = r^2\\\\Take \ square \ root \ on \ both \ sides\\\\\frac{1}{2} = r\\\\r = \frac{1}{2}[/tex]

Now, to find the second term, we will have to multiply r with the first term.

So,

g₂ = (24) × (1/2)

g₂ = 12

[tex]\rule[225]{224}{2}[/tex]

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