Question

A rope is swinging in such a way that the length of the arc is decreasing geometrically. If the the first arc is 18 feet long and the third arc is 8 feet long, what is the length of the second arc?

Explain step by step.
Geometric Sequence:
In mathematics, a sequence in which each number is multiplied by its previous term is called a geometric sequence.
The standard form of the geometric sequence is:
an=a1×rn−1Where, r = Common ratio a1= First term an=n th term

Answer 1

To find the length of the second arc, we can determine the common ratio by using the lengths of the first and third arcs in the geometric sequence formula. Then, we apply the common ratio to calculate the length of the second arc.

Step-by-step explanation:

The question asks for the length of the second arc in a geometric sequence where the first arc length is 18 feet, and the third arc length is 8 feet. To find the second arc length, we can use the formula for a geometric sequence, which is an = a1 × r^(n-1), where a1 is the first term, an is the nth term, and r is the common ratio.

Since we know the first term (a1) is 18 feet and the third term (a3) is 8 feet, we can use these values to find the common ratio. The third term formula would be a3 = a1 × r^(3-1), which simplifies to 8 = 18 × r^2. Solving for r, we divide both sides by 18 to get r^2 = 8/18, and then take the square root of both sides to find the common ratio r.

Once the common ratio is found, we can calculate the second term (a2) using the formula a2 = a1 × r^(2-1) which simplifies to a2 = a1 × r. Through this process, we can determine the length of the second arc in the given geometric sequence.

Answer 2

The length of the second arc is 12 feet.

Step-by-step explanation:

A geometric sequence is a sequence in which each number is multiplied by its previous term. In this case, the arc lengths are decreasing geometrically. Let's denote the length of the first arc as a1 and the length of the third arc as a3.

The standard form of a geometric sequence is an = a1 * rn-1. Since a1 is given as 18 feet and a3 is given as 8 feet, we can plug these values into the equation to solve for the common ratio r:

8 = 18 * r2

Dividing both sides by 18 gives:

r2 = 8/18

Simplifying further:

r2 = 4/9

Taking the square root of both sides:

r = ±2/3

Since geometric sequences can have both positive and negative common ratios, we consider both possibilities for r. Using the positive value, the length of the second arc can be calculated as:

a2 = a1 * r

a2 = 18 * 2/3

a2 = 12 feet