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4 years ago

In how many ways can six people sit in a six-passenger car?

Mathematics

1 answer:

nekit [7.7K]4 years ago

5 0

Answer:

number of ways = 720

Step-by-step explanation:

The number of ways six people sit in a six-passenger car is given by the number of permutations of 6 elements in 6 different positions ( seats), then

number of ways = number of permutations of 6 elements = 6! = 6 * 5 * 4 *3 * 2 * 1 = 720

Since the first person that sits can be on any of the seats , but then the second person that sits can choose any of 5 seats (since the first person had already occupied one) , the third can choose 4 ... and so on.

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A cylinder and a cone have congruent heights and radii.what is the ratio of the volume of the cone to the volume of the cylinder

NemiM [27]

Answer:

1: 3 ratio of the volume of the cone to the volume of the cylinder

Step-by-step explanation:

Volume of cone(V) is given by:

$V = \frac{1}{3} \pi r^2h$

where, r is the radius and h is the height of the cone.

Volume of cylinder(V') is given by:

$V' = \pi r'^2h'$

where, r' is the radius and h' is the height of the cylinder.

As per the statement:

A cylinder and a cone have congruent heights and radii.

⇒r = r' and h = h'

then;

$\frac{V}{V'} = \frac{ \frac{1}{3} \pi r^2h}{\pi r'^2h'} = \frac{ \frac{1}{3} \pi r'^2h'}{\pi r'^2h'}$

⇒ $\frac{V}{V'} =\frac{1}{3} = 1 : 3$

Therefore, the ratio of the volume of the cone to the volume of the cylinder is, 1 : 3

4 0

3 years ago

Read 2 more answers

Round off the following numbers to the nearest thousand. 19. 674,620 20. 355,500

Oksana_A [137]

Answer:

1) 675,000.

2) 356,000

Step-by-step explanation:

There are some rules to round off to the nearest thousand.

1. While rounding off to the nearest thousand, if the digit in the hundreds place is between 0 – 4 i.e., < 5, then the hundreds place is replaced by ‘0’.

2. If the digit in the hundreds place is = to or > 5, then the hundreds place is replaced by ‘0’ and the thousands place is increased by 1.

We have given two numerals:

1)<u> 674,620</u>

According to the second rule defined the digit at the hundreds place is greater than 5 than the thousands place will be increased by 1.

The answer is 675,000.

2) <u>355,500</u>

According to the second rule defined the digit at the hundreds place is greater than 5 than the thousands place will be increased by 1.

The answer is 356,000....

4 0

3 years ago

Norman can type 44 words in a minute. At this rate, how many words can he type in 6 minutes?

inysia [295]

Answer:

264

Step-by-step explanation:

8 0

3 years ago

Read 2 more answers

Simplify: 19w5+ (-3075)

AleksAgata [21]

Simplified: 19w^5 - 3075

7 0

3 years ago

I'm having trouble with #2. I've got it down to the part where it would be the integral of 5cos^3(pheta)/sin(pheta). I'm not sur

Butoxors [25]

$\displaystyle\int\frac{\sqrt{25-x^2}}x\,\mathrm dx$

Setting $x=5\sin\theta$ , you have $\mathrm dx=5\cos\theta\,\mathrm d\theta$ . Then the integral becomes

$\displaystyle\int\frac{\sqrt{25-(5\sin\theta)^2}}{5\sin\theta}5\cos\theta\,\mathrm d\theta$
$\displaystyle\int\sqrt{25-25\sin^2\theta}\dfrac{\cos\theta}{\sin\theta}\,\mathrm d\theta$
$\displaystyle5\int\sqrt{1-\sin^2\theta}\dfrac{\cos\theta}{\sin\theta}\,\mathrm d\theta$
$\displaystyle5\int\sqrt{\cos^2\theta}\dfrac{\cos\theta}{\sin\theta}\,\mathrm d\theta$

Now, $\sqrt{x^2}=|x|$ in general. But since we want our substitution $x=5\sin\theta$ to be invertible, we are tacitly assuming that we're working over a restricted domain. In particular, this means $\theta=\sin^{-1}\dfrac x5$ , which implies that $\left|\dfrac x5\right|\le1$ , or equivalently that $|\theta|\le\dfrac\pi2$ . Over this domain, $\cos\theta\ge0$ , so $\sqrt{\cos^2\theta}=|\cos\theta|=\cos\theta$ .

Long story short, this allows us to go from

$\displaystyle5\int\sqrt{\cos^2\theta}\dfrac{\cos\theta}{\sin\theta}\,\mathrm d\theta$

to

$\displaystyle5\int\cos\theta\dfrac{\cos\theta}{\sin\theta}\,\mathrm d\theta$
$\displaystyle5\int\dfrac{\cos^2\theta}{\sin\theta}\,\mathrm d\theta$

Computing the remaining integral isn't difficult. Expand the numerator with the Pythagorean identity to get

$\dfrac{\cos^2\theta}{\sin\theta}=\dfrac{1-\sin^2\theta}{\sin\theta}=\csc\theta-\sin\theta$

Then integrate term-by-term to get

$\displaystyle5\left(\int\csc\theta\,\mathrm d\theta-\int\sin\theta\,\mathrm d\theta\right)$
$=-5\ln|\csc\theta+\cot\theta|+\cos\theta+C$

Now undo the substitution to get the antiderivative back in terms of $x$ .

$=-5\ln\left|\csc\left(\sin^{-1}\dfrac x5\right)+\cot\left(\sin^{-1}\dfrac x5\right)\right|+\cos\left(\sin^{-1}\dfrac x5\right)+C$

and using basic trigonometric properties (e.g. Pythagorean theorem) this reduces to

$=-5\ln\left|\dfrac{5+\sqrt{25-x^2}}x\right|+\sqrt{25-x^2}+C$

4 0

3 years ago

Read 2 more answers