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3 years ago

The value of y varies directly with x and the constant of variation is 4. What is the value of x when y=84?

Mathematics

1 answer:

anygoal [31]3 years ago

6 0

Y varying directly with x means y equals a constant multiplying x: y=kx
in this case, k=4, so 84=4x
x=21

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I found the interval of convergence, but I am not sure how to do the second part, finding the sum of the series as a function of

antiseptic1488 [7]

The given series is geometric with common ratio $6^x - 9$ , which converges if $|6^x - 9|$ (i.e. the interval of convergence). We have the well-known result

$\displaystyle |r| < 1 \implies \sum_{n=0}^\infty ar^n = \frac{a}{1-r}$

If you're not familiar with that result, it's easy to reproduce.

Let $S_N$ be the $N$ -th partial sum of the infinite series,

$\displaystyle S_N = \sum_{n=0}^N \left(6^x - 9\right)^n = 1 + \left(6^x - 9\right) + \left(6^x - 9\right)^2 + \cdots + \left(6^x - 9\right)^N$

Multiply both sides by the ratio.

$\left(6^x - 9\right) S_N = \left(6^x - 9\right) + \left(6^x - 9\right)^2 + \left(6^x - 9\right)^3 + \cdots + \left(6^x - 9\right)^{N+1}$

Subtract this from $S_N$ to eliminate all the powers of the ratio between 0 and $N+1$ .

$\left(1 - \left(6^x - 9\right)\right) S_N = 1 - \left(6^x - 9\right)^{N+1}$

Solve for $S_N$ .

$S_N = \dfrac{1 - \left(6^x - 9\right)^{N+1}}{10-6^x}$

Now as $N\to\infty$ , the exponential term converges to 0 and we're left with

$\displaystyle \sum_{n=0}^\infty \left(6^x-9\right)^n = \lim_{N\to\infty} S_N = \boxed{\frac1{10-6^x}}$

7 0

2 years ago

Which is the best estimate of 174% of 141?

Mila [183]

Yes..the absolute answer is 245.34

3 0

3 years ago

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Which of the following values is not in the domain of y = tanx?

SSSSS [86.1K]

Answer:

I think its the first option

Step-by-step explanation:

I'm not sure

7 0

3 years ago

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Which expression uses the greatest common factor and the distributive property to rewrite the sum 7 + 75

Lady_Fox [76]

Answer:

$75 + 45$ $= 15(3 + 5)$

Step-by-step explanation:

Given

$75 + 45$ -- Corrected

Required

Use the GCF to write it as a distributive property

First, we find the GCF of 75 and 45

$45 = 3 * 3 * 5$

$75 = 3 * 5 * 5$

The common factor is:

$GCF = 3 * 5$

$GCF = 15$

So, $75 + 45$ becomes

$75 + 45$ $= 15 * 3 + 15 * 5$

Factorize

$75 + 45$ $= 15(3 + 5)$

Hence, the sum has been rewritten as $15(3 + 5)$

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3 years ago

Find the rectangular coordinates of the point with the polar coordinates (8, 3 divided by 2 pi).

ludmilkaskok [199]

(r, theta)= (8, 3/2 pi)
r=(x^2 +y^2)^(1/2)
theta= 3/2 pi
x= r(costheta)
y=r(sintheta)
x=8(cos(3/2 pi))
y=8(sin(3/2 pi))
x=8(0)
y=8(-1)
x=0
y=-8
r=((-8)^2+(0)^2)^(1/2)
r=(64+0)^(1/2)
r=8
rectangular coordinates= (0,-8)

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4 years ago

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