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1 year ago

5

a point in space $(x,y,z)$ is randomly selected so that $-1\le x \le 1$,$-1\le y \le 1$,$-1\le z \le 1$. what is the probability

that $x^2 y^2 z^2\le 1$?

1 answer:

Yuliya22 [10]1 year ago

5 0

Probability that $x^2+y^2+z^2\leq 1$ is 0.523

Given that a point (x, y, z) is in space which is randomly selected as

$-1\le x \le 1$,$-1\le y \le 1$,$-1\le z \le 1$$

These three equations represents a cube in space.

Length of each side of the cube = 1 - -1 = 2

Now $x^2+y^2+z^2\leq 1$ represents a sphere with center (0, 0, 0) and of radius less than or equal to 1.

So we can imagine that this sphere is inside the cube.

So the probability that $x^2+y^2+z^2\leq 1$ = Volume of the sphere/Volume of the cube

Volume of sphere with radius 1 = $\frac{4}{3} \pi r^3=\frac{4}{3} \pi \times 1 = \frac{4}{3}\times 3.14 = 4.186$ cubic unit

Volume of the cube of length 2 = $2^3 = 8$ cubic units

So the probability that $x^2+y^2+z^2\leq 1$ = 4.186/8 = 0.523

Learn more about probability at brainly.com/question/25870256

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Answer:

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Step-by-step explanation:

I think what it is trying to say is that it wants the solution to multiplying all of those. The (...) simply means that it wants you to continue that pattern in what you are supposed to be multiplying, but stop at $\frac{24}{25}$ .

That would mean you are technically supposed to be multiplying:

$\frac{1}{2} \frac{2}{3} \frac{3}{4} \frac{4}{5} \frac{5}{6} \frac{6}{7} \frac{7}{8} \frac{8}{9} \frac{9}{10} \frac{10}{11} \frac{11}{12} \frac{12}{13} \frac{13}{14} \frac{14}{15} \frac{15}{16} \frac{16}{17} \frac{17}{18} \frac{18}{19} \frac{19}{20} \frac{20}{21} \frac{21}{22} \frac{22}{23} \frac{23}{24} \frac{24}{25}$

That is a lot and unfortunately, none of the individual fractions can be simplified within that. The final answer would be able to be simplified, though.

Multiplying the first four shown: $\frac{1}{2}\frac{2}{3} \frac{3}{4} \frac{4}{5}$ you end up with $\frac{24}{120}$ . Both the numerator (top) and the denominator (bottom) are divisible by 24. Dividing top and bottom would simplify this to $\frac{1}{5}$ .

Now, let's take the next four.

$\frac{5}{6}\frac{6}{7} \frac{7}{8} \frac{8}{9}$ allows you to end up with $\frac{1680}{3024}$ . Both the numerator (top) and the denominator (bottom) are divisible by 336. You are left with $\frac{5}{9}$ .

Now, let's take the next four.

$\frac{9}{10}\frac{10}{11} \frac{11}{12} \frac{12}{13}$ . Multiplying these gives you $\frac{11880}{17160}$ . Both the numerator (top) and the denominator (bottom) are divisible by 1320. You are left with $\frac{9}{13}$ .

Now, let's take the next four.

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The next four are $\frac{17}{18}\frac{18}{19} \frac{19}{20} \frac{20}{21}$ . Multiplying these, you are left with $\frac{116280}{143640}$ . Both the numerator (top) and the denominator (bottom) are divisible by 6840. We are left with $\frac{17}{21}$ . This is the exact guess I had made when following the pattern, and so the next one is most likely going to be the other guess as well.

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Simplified, your final answer is: $\frac{1}{25}$ .

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$▪▪▪▪▪▪▪▪▪▪▪▪▪ {\huge\mathfrak{Answer}}▪▪▪▪▪▪▪▪▪▪▪▪▪▪$

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3 years ago

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