1. determine the value of x for this right triangle.\n34 in\nx\n52 in\n2. estimate how many times smaller…

Explanation:

Step1: Apply Pythagorean theorem for right - triangle

For a right - triangle with sides $a = 34$ in, $b = 52$ in and hypotenuse $x$, by the Pythagorean theorem $x=\sqrt{a^{2}+b^{2}}=\sqrt{34^{2}+52^{2}}=\sqrt{1156 + 2704}=\sqrt{3860}\approx62.1$ in.

Step2: Calculate mass ratio

To find how many times smaller the mass of an atom is compared to the mass of an elephant, divide the mass of the atom by the mass of the elephant. Let $m_{atom}=1.9944\times 10^{- 26}$ kg and $m_{elephant}=2\times10^{3}$ kg. The ratio is $\frac{m_{atom}}{m_{elephant}}=\frac{1.9944\times 10^{-26}}{2\times10^{3}}=0.9972\times10^{-29}=9.972\times 10^{-30}$.

Step3: Approximate the value of $8\pi$

We know that $\pi\approx3.14$, so $8\pi\approx8\times3.14 = 25.12$.

Step4: Reflect points about the y - axis

The rule for reflecting a point $(x,y)$ about the y - axis is $(-x,y)$. For point $A(2,5)$, its reflection $A'(-2,5)$. For point $B(8,5)$, its reflection $B'(-8,5)$. For point $C(3,-4)$, its reflection $C'(-3,-4)$.

Step5: Analyze scatter - plot

If the y - axis shows grades and the x - axis represents hours studying, we can gather the relationship between the number of hours studied and the grades achieved. Generally, if the points show an upward trend, it means that as the number of hours studying increases, the grades tend to increase. If the points are scattered randomly, there may be little to no linear relationship between hours studying and grades.

Answer:

1. $x\approx62.1$ in
2. The mass of an atom is $9.972\times 10^{-30}$ times the mass of an elephant.
3. $8\pi\approx25.12$
4. $A'(-2,5),B'(-8,5),C'(-3,-4)$
5. Relationship between hours studying and grades (upward trend may indicate positive correlation, random scatter may indicate no linear correlation).