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4 years ago

Write the equation of the sphere in standard form. x2 + y2 + z2 + 8x − 6y + 4z + 20 = 0

Mathematics

1 answer:

Nuetrik [128]4 years ago

5 0

Answer:

$(x+4)^2 + (y-3)^2 + (z+2)^2 - 3^2 = 0$

Step-by-step explanation:

You have to find a, b, c, r so that the polynomial expands to match the given equation.

$(x-a)^2 + (y-b)^2 + (z-c)^2 - r^2 =\\x^2-2ax+a^2+\\y^2-2by+b^2+\\z^2-2cz+c^2-r^2 =\\ x^2 + y^2 + z^2 + 8x - 6y + 4z + 20$

$(a^2+b^2+c^2-r^2-20)+\\x(-8-2a)+y(6-2b)+z(-4-2c)=0$

$-8-2a=0, 6-2b=0, -4-2c=0\\a=-4, b=3, c=-2$

$(-4)^2+3^2+(-2)^2-r^2-20 = 0$

$16+9+4-20=r^2$

$r^2=9$

$r=3$

$(x+4)^2 + (y-3)^2 + (z+2)^2 - 3^2 = 0$

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95.44% probability that the sample proportion will differ from the population proportion by less than 0.03.

Step-by-step explanation:

To solve this question, we need to understand the normal probability distribution and the central limit theorem.

Normal probability distribution

When the distribution is normal, we use the z-score formula.

In a set with mean $\mu$ and standard deviation $\sigma$ , the zscore of a measure X is given by:

$Z = \frac{X - \mu}{\sigma}$

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.

Central Limit Theorem

The Central Limit Theorem estabilishes that, for a normally distributed random variable X, with mean $\mu$ and standard deviation $\sigma$ , the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean $\mu$ and standard deviation $s = \frac{\sigma}{\sqrt{n}}$ .

For a skewed variable, the Central Limit Theorem can also be applied, as long as n is at least 30.

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