Mathematics, 03.08.2019 12:20, LexyClaire

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.

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Answers

The correct answer is 9.8 ft.

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

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.

answered: rydersasser12

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

$a_{n}=a(r)^{n-1}$

where $a_{n}$ 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 ⇒ $a_{3}=a(r)^{2}=18$ ------(1)

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

Now we divide equation 2 from equation 2

$\frac{a_{7}}{a_{3} }=\frac{a.r^{6}}{a.r^{2}}=\frac{8}{18}$

we solve it further

$r^{4}=\frac{4}{9}$

$r^{2}=\sqrt{\frac{4}{9}}=\frac{2}{3}$

$r=\sqrt{\frac{2}{3}}=\sqrt{0.667}=0.817$

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

$a_{6}=(26.99).(0.0.817)^{6-1}=(26.99)(0.817)^{5}=(26.99).(0.363)=9.80$

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

answered: Guest

step-by-step explanation:

answered: Guest

option: d is the correct answer.

d) none of the above.

step-by-step explanation:

clearly from the figure attached to the question we could see that there is no shape given. hence without any shape given in the figure and after rotation we can't obtained a square pyramid.

since, to obtain a image we need a proper figure of the pre-image so that it can be transformed.

hence, the correct answer is:

d) none of the above.

step-by-step explanation: