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PLEASE HELP! I'M ON A TIMER
A rope is swinging in such a way that the length of the arc traced by a knot at its bottom end is decreasing geometrically. If the third arc is 18 ft. long and the seventh arc is 8 ft. long, what is the length of the arc on the sixth swing? Round your answer to the nearest tenth of a foot.

Respuesta :

The correct answer is 9.8 ft.

Explanation:
This is a geometric sequence, which follows the explicit formula 
[tex]g_n=g_1\times r^{n-1}[/tex] 

where g₁ is the first term, r is the common ratio and n is the term number.

We know that the third term is 18; this gives us 18=g₁×r³⁻¹ or 18=g₁×r².

We also know the seventh term is 8, which gives us 8=g₁×r⁷⁻¹ or 8=g₁×r⁶.

Solving for g₁ in the third term gives us g₁=18/r², and solving for g₁ in the seventh term gives us g₁=8/r⁶. They both equal g₁ so we set them equal to each other:

18/r² = 8/r⁶.
Multiply both sides by r⁶, which gives us
18r⁶/r² = 8.

Using our properties of exponents, we have 18r⁴ = 8. Divide both sides by 18, which gives us
r⁴=8/18.

We can find the fourth root by taking the square root twice:
taking the square root gives us r² = √8/√18.

Simplifying √8 we get 2√2, and simplifying √18 gives us 3√2; we now have

r²=2√2)/3√2.

The √2 will cancel, leaving r²=2/3. Taking the square root again, we have

r=√2/√3; simplifying this gives us r=√6/3.

We can now work backward to find the sixth term using the seventh one; Divide 8 by √6/3. Dividing by a fraction means multiplying by the reciprocal, so we multiply 8 by 3/√6; this gives us 24/√6, and in a calculator that gives us 9.8 ft.

Answer:

Length of the arc on the sixth swing = 9.80 ft

Step-by-step explanation:

A rope is swinging in such a way that the length of the arc traced by a knot at its bottom end is decreasing geometrically.

Length of 3rd arc = 18 ft

Length of 7th arc = 8 ft

We have to find the length of arc formed in a the 6th swing.

As we know in a geometric sequence, explicit formula is given as

[tex]a_{n}=a(r)^{n-1}[/tex]

where [tex]a_{n}[/tex] is the nth term, a is the first term, r is the common ratio and n is the number of term

Now for 3rd term of the sequence ⇒ [tex]a_{3}=a(r)^{2}=18[/tex]------(1)

For 7th term of the sequence ⇒ [tex]a_{7}=a(r)^{7-1}=ar^{6}=8[/tex] ------(2)

Now we divide equation 2 from equation 2

[tex]\frac{a_{7}}{a_{3} }=\frac{a.r^{6}}{a.r^{2}}=\frac{8}{18}[/tex]

we solve it further

[tex]r^{4}=\frac{4}{9}[/tex]

[tex]r^{2}=\sqrt{\frac{4}{9}}=\frac{2}{3}[/tex]

[tex]r=\sqrt{\frac{2}{3}}=\sqrt{0.667}=0.817[/tex]

Now we put the value of r in equation 1

a.r² = 18

a.(√0.667)²= 18

a×0.667 = 18 ⇒ a = 26.986

Now we will calculate the 6th term of this sequence

[tex]a_{6}=(26.99).(0.0.817)^{6-1}=(26.99)(0.817)^{5}=(26.99).(0.363)=9.80[/tex]

Answer is Length of the arc on the 6th swing = 9.80 ft

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