What are the values the values of a and b?

Mathematics

1 answer:

Setler [38]3 years ago

4 0

The segment marked 20 is the geometric mean of the two pieces into which it splits the hypotenuse. You get the proportion

$\frac{20}{a} = \frac{21}{20}$

Solving,

$21a =(20)(20) \\ 21a = 400 \\ a = \frac{400}{21}$

To find b, use the fact that b is the geometric mean of the whole hypotenuse, 21 + a, and the part that is attached to it, a.

$\frac{b}{a} = \frac{21+a}{b}$

A little arithmetic shows that $21+a = \frac{841}{21}$ .

$\frac{b}{\frac{400}{21}} = \frac{\frac{841}{21}}{b} \\ b^2 = \frac{336400}{441} \\ b=\frac{580}{21}$

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$1\leq x\leq 3,\quad 1\leq y \leq 5$

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Now we use the assumption of independence to claim that

$P(X=x, Y=y) = P(X=x)\cdot P(Y=y) =\dfrac{1}{3}\cdot\dfrac{1}{5} = \dfrac{1}{15}$

Now, we simply have to count in how many ways we can obtain every possible outcome for the sum. Consider the attached table: we can see that we can obtain:

2 in a unique way (1+1)
3 in two possible ways (1+2, 2+1)
4 in three possible ways
5 in three possible ways
6 in three possible ways
7 in two possible ways
8 in a unique way

This implies that the probabilities of the outcomes of $W=X+Y$ are the number of possible ways divided by 15: we can obtain 2 and 8 with probability 1/15, 3 and 7 with probability 2/15, and 4, 5 and 6 with probabilities 3/15=1/5