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3 years ago

Which is the MOST reasonable estimate for "the square root of eight divided by two to the second power?

Mathematics

1 answer:

neonofarm [45]3 years ago

6 0

Answer:

We want to estimate:

$\frac{\sqrt{8} }{2^2}$

We can rewrite this as:

$\frac{\sqrt{8} }{2^2} = \frac{\sqrt{8} }{4} =\frac{\sqrt{8} }{\sqrt{4^2} } =\sqrt{\frac{8}{4*4} } = \sqrt{\frac{2}{4} } = \frac{1}{\sqrt{2} }$

And √2 is a notable value, we know that:

√2 ≈ 1.41

Then:

$\frac{1}{\sqrt{2} } = \frac{1}{1.41} = 0.7$

then:

$\frac{\sqrt{8} }{2^2} = 0.7$

is a really good estimation.

if instead:

"the square root of eight divided by two to the second power"

refers to:

$\sqrt{\frac{8}{2^2} }$

is easier, we just can replace 2^2 = 4, then we get:

$\sqrt{\frac{8}{2^2} } = \sqrt{\frac{8}{4} } = \sqrt{2} = 1.41$

Is also a really good aproximation.

You might be interested in

The age of United States Presidents on the day of their first inauguration follows a Normal distribution with mean 56 and standa

Talja [164]

Answer:

a) 0.7088 = 70.88% probability that a randomly selected President was less than 60 years old on the day of their first inauguration.

b) The 75th percentile for the age of United States Presidents on the day of inauguration is 61.

c) 0.8643 = 86.43% probability that the average age on the day of their first inauguration for a random sample of 4 United States Presidents exceeds 60 years.

Step-by-step explanation:

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

Normal Probability Distribution:

Problems of normal distributions can be solved using 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.

The age of United States Presidents on the day of their first inauguration follows a Normal distribution with mean 56 and standard deviation 7.3.

This means that $\mu = 56, \sigma = 7.3$

(a) (5 points) Compute the probability that a randomly selected President was less than 60 years old on the day of their first inauguration.

This is the pvalue of Z when X = 60. So

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

$Z = \frac{60 - 56}{7.3}$

$Z = 0.55$

$Z = 0.55$ has a pvalue of 0.7088

0.7088 = 70.88% probability that a randomly selected President was less than 60 years old on the day of their first inauguration.

(b) (5 points) Compute the 75th percentile for the age of United States Presidents on the day of inauguration.

This is X when Z has a pvalue of 0.75. So X when Z = 0.675.

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

$0.675 = \frac{X - 56}{7.3}$

$X - 56 = 0.675*7.3$

$X = 61$

The 75th percentile for the age of United States Presidents on the day of inauguration is 61.

(c) (5 points) Compute the probability that the average age on the day of their first inauguration for a random sample of 4 United States Presidents exceeds 60 years.

Now, by the Central Limit Theorem, we have that $n = 4, s = \frac{7.3}{\sqrt{4}} = 3.65$

This is the pvalue of Z when X = 60. So

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

By the Central Limit Theorem

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

$Z = \frac{60 - 56}{3.65}$

$Z = 1.1$

$Z = 1.1$ has a pvalue of 0.8643

0.8643 = 86.43% probability that the average age on the day of their first inauguration for a random sample of 4 United States Presidents exceeds 60 years.

4 0

3 years ago

When an inequality is in slope-intercept form, what is the form?*

horsena [70]

Answer:

y=mx+b

Step-by-step explanation:

this is slope intercept form but you must fill in the letters with numbers

3 0

3 years ago

A car company says that the mean gas mileage for its luxury sedan is at least 24 miles per gallon (mpg). You believe the claim

Anastaziya [24]

Answer:

$t=\frac{23-24}{\frac{5}{\sqrt{5}}}=-0.447$

$p_v =P(t_{(4)}$

If we compare the p value and the significance level given $\alpha=0.1$ we see that $p_v>\alpha$ so we can conclude that we have enough evidence to fail reject the null hypothesis, so we can't reject the claim that the true mean is at least 24 mpg (null hypothesis) at 1% of signficance.

Step-by-step explanation:

Data given and notation

$\bar X=23$ represent the sample mean

$s=5$ represent the sample standard deviation for the sample

$n=5$ sample size

$\mu_o =24$ represent the value that we want to test

$\alpha=0.1$ represent the significance level for the hypothesis test.

t would represent the statistic (variable of interest)

$p_v$ represent the p value for the test (variable of interest)

State the null and alternative hypotheses.

We need to conduct a hypothesis in order to check if the mean is lower than 24 (rejecting the claim proposed), the system of hypothesis would be:

Null hypothesis: $\mu \geq 24$

Alternative hypothesis: $\mu < 24$

If we analyze the size for the sample is < 30 and we don't know the population deviation so is better apply a t test to compare the actual mean to the reference value, and the statistic is given by:

$t=\frac{\bar X-\mu_o}{\frac{s}{\sqrt{n}}}$ (1)

t-test: "Is used to compare group means. Is one of the most common tests and is used to determine if the mean is (higher, less or not equal) to an specified value".

Calculate the statistic

We can replace in formula (1) the info given like this:

$t=\frac{23-24}{\frac{5}{\sqrt{5}}}=-0.447$

P-value

The first step is calculate the degrees of freedom, on this case:

$df=n-1=5-1=4$

Since is a one sided test the p value would be:

$p_v =P(t_{(4)}$

Conclusion

5 0

3 years ago

Approximate cos (12π/ 13) by using a linear approximation with f (x) = cos x.

Katarina [22]

First pick a value of $x$ close to $\dfrac{12\pi}{13}$ . You should be fine with $x=\pi$ .

The linear approximation of $f(x)$ at $x=c$ is given by

$f(c)\approx f(a)+f'(a)(c-a)$

where $x=a$ is some fixed value close to $x=c$ . You have

$f(x)=\cos x\implies f'(x)=-\sin x$

so

$f\left(\dfrac{12\pi}{13}\right)\approx f(\pi)+f'(\pi)\left(\dfrac{12\pi}{13}-\pi\right)$
$\cos\dfrac{12\pi}{13}\approx\cos\pi-\sin(\pi)\left(-\dfrac\pi{13}\right)$
$\cos\dfrac{12\pi}{13}\approx-1$

The actual value is closer to -0.9709, so the approximation is decent.

4 0

3 years ago

A high school football game Jill buys 6 hot dogs and 4 soft drinks for

ryzh [129]

Answer: D

Step-by-step explanation:Lets label hot dogs as 'h' and soft drinks as 's'

6h + 4s = 13 (Equation 1)

3h + 4s = 8.50 (Equation 2)

We can solve by process of elimination.

Subtract equation 1 from equation 2.

We get:

3h = 4.5

h = 1.5

So that means that the hot dog costs $1.50 which is answer choice D.

7 0

3 years ago