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natta225 [31]
3 years ago
5

PLEASE HELP! I'M ON A TIMER

Mathematics
2 answers:
Marysya12 [62]3 years ago
5 0

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

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

marta [7]3 years ago
3 0
<span>The correct answer is 9.8 ft.

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

where g</span>₁<span> 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=g</span>₁×<span>r</span>³⁻¹<span> or 18=g</span>₁×<span>r</span>²<span>.

We also know the seventh term is 8, which gives us 8=g</span>₁×<span>r</span>⁷⁻¹<span> or 8=g</span>₁×<span>r</span>⁶<span>.

Solving for g</span>₁<span> in the third term gives us g</span>₁<span>=18/r</span>²<span>, and solving for g</span>₁<span> in the seventh term gives us g</span>₁<span>=8/r</span>⁶<span>. They both equal g</span>₁<span> so we set them equal to each other:

18/r</span>²<span> = 8/r</span>⁶<span>.
Multiply both sides by r</span>⁶<span>, which gives us
18r</span>⁶<span>/r</span>²<span> = 8.

Using our properties of exponents, we have 18r</span>⁴<span> = 8. Divide both sides by 18, which gives us
r</span>⁴<span>=8/18.

We can find the fourth root by taking the square root twice:
taking the square root gives us r</span>²<span> = </span>√<span>8/</span>√<span>18.

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

r</span>²<span>=2</span>√<span>2)/3</span>√<span>2.

The </span>√<span>2 will cancel, leaving r</span>²<span>=2/3. Taking the square root again, we have

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

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