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3 years ago

Solve the proportional equation 3/r = 5/r+3

Mathematics

1 answer:

romanna [79]3 years ago

6 0

Answer:

r=9/2 or 4.5

Step-by-step explanation:

First of all our goal is to get R by itself

by cross multiplying we get,

(r+3)*3=5r

by distributive property we get

3r+9=5r

substracting 3r from both sides we get

9=2r

by dividing by 2 in both sides we get

r=9/2 or 4.5

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you roll a six sided number cube and flip a coin what is the probability of rolling a number greater than 5 and flipping heads?

Whitepunk [10]

Answer:

0.08333... or 8.333...%

Step-by-step explanation:

1/6 * 1/2 = 1/12

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3 years ago

Read 2 more answers

Determine the above sequence converges or diverges. If the sequence converges determine its limit

marshall27 [118]

Answer:

This series is convergent. The partial sums of this series converge to $\displaystyle \frac{2}{3}$ .

Step-by-step explanation:

The $n$ th partial sum of a series is the sum of its first $n\!\!$ terms. In symbols, if $a_n$ denote the $n\!$ th term of the original series, the $\! n$ th partial sum of this series would be:

$\begin{aligned} S_n &= \sum\limits_{k = 1}^{n} a_k \\ &= a_1 + a_2 + \cdots + a_{k}\end{aligned}$ .

A series is convergent if the limit of its partial sums, $\displaystyle \lim\limits_{n \to \infty} S_{n}$ , exists (should be a finite number.)

In this question, the $n$ th term of this original series is:

$\displaystyle a_{n} = \frac{{(-1)}^{n+1}}{{2}^{n}}$ .

The first thing to notice is the ${(-1)}^{n+1}$ in the expression for the $n$ th term of this series. Because of this expression, signs of consecutive terms of this series would alternate between positive and negative. This series is considered an alternating series.

One useful property of alternating series is that it would be relatively easy to find out if the series is convergent (in other words, whether $\displaystyle \lim\limits_{n \to \infty} S_{n}$ exists.)

If $\lbrace a_n \rbrace$ is an alternating series (signs of consecutive terms alternate,) it would be convergent (that is: the partial sum limit $\displaystyle \lim\limits_{n \to \infty} S_{n}$ exists) as long as $\lim\limits_{n \to \infty} |a_{n}| = 0$ .

For the alternating series in this question, indeed:

$\begin{aligned}\lim\limits_{n \to \infty} |a_n| &= \lim\limits_{n \to \infty} \left|\frac{{(-1)}^{n+1}}{{2}^{n}}\right| = \lim\limits_{n \to \infty} {\left(\frac{1}{2}\right)}^{n} =0\end{aligned}$ .

Therefore, this series is indeed convergent. However, this conclusion doesn't give the exact value of $\displaystyle \lim\limits_{n \to \infty} S_{n}$ . The exact value of that limit needs to be found in other ways.

Notice that $\lbrace a_n \rbrace$ is a geometric series with the first term is $a_0 = (-1)$ while the common ratio is $r = (- 1/ 2)$ . Apply the formula for the sum of geometric series to find an expression for $S_n$ :

$\begin{aligned}S_n &= \frac{a_0 \cdot \left(1 - r^{n}\right)}{1 - r} \\ &= \frac{\displaystyle (-1) \cdot \left(1 - {(-1 / 2)}^{n}\right)}{1 - (-1/2)} \\ &= \frac{-1 + {(-1 / 2)}^{n}}{3/2} = -\frac{2}{3} + \frac{2}{3} \cdot {\left(-\frac{1}{2}\right)}^{n}\end{aligned}$ .

Evaluate the limit $\displaystyle \lim\limits_{n \to \infty} S_{n}$ :

$\begin{aligned} \lim\limits_{n \to \infty} S_{n} &= \lim\limits_{n \to \infty} \left(-\frac{2}{3} + \frac{2}{3} \cdot {\left(-\frac{1}{2}\right)}^{n}\right) \\ &= -\frac{2}{3} + \frac{2}{3} \cdot \underbrace{\lim\limits_{n \to \infty} \left[{\left(-\frac{1}{2}\right)}^{n} \right] }_{0}= -\frac{2}{3}\end{aligned}}_$ .

Therefore, the partial sum of this series converges to $\displaystyle \left(- \frac{2}{3}\right)$ .

8 0

3 years ago

Pythagoras was born about 582 bc. Isaac Newton was born in 1643 ad. How many years apart were they born?

malfutka [58]

B.C. counts down to 0, and that's when A.D. starts counting up. That means you can add 582 and 1643, to get their difference. Pythagoras and Newton were born 2,225 years apart.

8 0

4 years ago

A hotel reservation number consists of 5 digits, followed by 1 letters, followed by 3 digits. How many different reservation num

sasho [114]

There are 10 possible digits and 26 possible letters.

To find the answer to this problem we have to multiply 9 as many digits has the hotel reservation number and 26 as many letters it has.

It means:

$9\cdot9\cdot9\cdot9\cdot9\cdot26\cdot9\cdot9\cdot9=1,119,214,746$

There are 1,119,214,746 possible reservation numbers.

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1 year ago

the first ferris wheel was built in 1893 in chicago. It’s diameter was 250 feet. How many feet did the ferris wheel rotate with

Mice21 [21]

Answer:

about 785.4

Step-by-step explanation:

The circumference of a circle is pi times the diameter:

C = πd

C = π·(250 ft) ≈ 785.4 ft

_____

Most scientific or graphing calculators have the value of π built in. If yours doesn't, a value good for at least 6 significant figures is the ratio 355/113.

5 0

3 years ago