What is the likelihood for the Fourth of July being on a Sunday? Use probability to explain your answer.

1 answer:

6 0

There are seven days in the week, and the 4th of July could be on any of them, so it has a 1/7 chance of being on a Sunday.

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Prove that if $w,z$ are complex numbers such that $|w|=|z|=1$ and $wz\ne -1$, then $\frac{w+z}{1+wz}$ is a real number.

mr_godi [17]

Answer:

See proof below

Step-by-step explanation:

Let $r=\frac{w+z}{1+wz}$ . If w=-z, then r=0 and r is real. Suppose that w≠-z, that is, r≠0.

Remember this useful identity: if x is a complex number then $x\bar{x}=|x|^2$ where $\bar{x}$ is the conjugate of x.

Now, using the properties of the conjugate (the conjugate of the sum(product) of two numbers is the sum(product) of the conjugates):

$\frac{r}{\bar{r}}=\frac{w+z}{1+wz} \left(\frac{1+\bar{w}\bar{z}}{\bar{w}+\bar{z}}{\right)$

= $\frac{(w+z)(1+\bar{w}\bar{z})}{(1+wz)(\bar{w}+\bar{z})}=\frac{w+z+w\bar{w}\bar{z}+z\bar{z}\bar{w}}{\bar{w}+\bar{z}+\bar{w}wz+\bar{z}zw}=\frac{w+z+w+|w|^2\bar{z}+|z|^2\bar{w}}{\bar{w}+\bar{z}+|w|^2z+|z|^2w}=\frac{w+z+\bar{z}+\bar{w}}{\bar{w}+\bar{z}+z+w}=1$

Thus $\frac{r}{\bar{r}}=1$ . From this, $r=\bar{r}$ . A complex number is real if and only if it is equal to its conjugate, therefore r is real.

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3. You drove 416 miles in 8<br>hours. How many miles did you<br>drive per hour?

daser333 [38]

Answer:

Step-by-step explanation:

You have 8 eight hours and 416miles, and you need to distribute those 416 miles to eight hours. so 416/8=52

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(4/2)^2*3.4*4

Andrej [43]

The original expression evaluates to 54.4 as shown in the first working.

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For some reason, 3.4 got transcribed to be 1.7 in the 3rd line of the second working. There is no math operation in the expression that requires 3.4 to be divided by 2.

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Help me!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

Masja [62]

Answer:

= 6/25 Is the correct answer.

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Aricela is trying to estimate the distance between her house and the park. On a map with a scale of 0.5 in. = 2 miles, the dista

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