Decide whether each of the following statements makes sense (or is clearly true) or does not make sense (or is clearly false). Explain your reasoning.

Someone wins the lottery every week, so I figure that if I keep playing eventually I will be the one who wins.

Find the arc length of the plane curve over the indicated interval.

Function: $y=x^2$, Interval: [0,3]

Decide whether each of the following statements makes sense (or is clearly true) or does not make sense (or is clearly false). Explain your reasoning.

Because either there is life on Mars or there is not, the probability of life on Mars is $0.5$.

(III) A $195\mathrm{-kg}$ projectile, fired with a speed of $116 \mathrm{~m} / \mathrm{s}$ at a $60.0^{\circ}$ angle, breaks into three pieces of equal mass at the highest point of its arc (where its velocity is horizontal). Two of the fragments move with the same speed right after the explosion as the entire projectile had just before the explosion; one of these moves vertically downward and the other horizontally. Determine $(b)$ the energy released in the explosion.

Ramira says that if two lines are perpendicular to the same line, then the two lines are parallel to each other. Is Ramira correct? If she is correct, complete the proof to justify the reasoning, or state why she is not correct.

Given: $\ell_1 \perp \ell_3 ; \ell_2 \perp \ell_3$

Prove: $\ell_1 \| \ell_2$

Calculate the double integral $\iint_{R} = x\sin(x + y)~dA, R = [0, \pi/6]\times[0, \pi/3]$.

Find the molarity of a 40.0% by mass aqueous solution of sulfuric acid, $\mathrm{H}_{2} \mathrm{SO}_{4}$, for which the density is 1.3057 g/mL.

Draw your own diagram and write "Given:" and "Prove:" statements in terms of your diagram. Do not write a proof. Given: A four-sided polygon with all four sides congruent (This figure is called a rhombus.) Conclusion: The lines joining opposite vertices are perpendicular.

Write an absolute value equation that has the given solution.

$0$

The star Mizar is actually a double star that only appears as two distinct stars when resolved by a telescope. Explain why.

Check whether each statement makes sense or does not make sense, and explain your reasoning. When using substitution to solve

$$\left\{\begin{array}{r} 5 x-4 y=9 \\ x-2 y=-3 \end{array}\right.$$

I find it easiest to solve for $x$ in the first equation.

Describe whether each of the following statements makes sense (or is clearly true) or does not make sense (or is clearly false). Explain your reasoning.

Sue prefers the Red Shuttle because it gets her to the airport in an hour and a half, while the Blue Shuttle takes $80$ minutes.

VALS makes sense to you? What do you think you like or dislike about it?

Even though larger circular gardens will fit in the grassy area, Lance decides that he would like the garden to have a diameter of 20 feet.

The landscape architect recommends surrounding the circular garden with decorative edging. The edging is sold in 12-foot sections that can bend into curves. How many sections of the edging will need to be purchased to surround the garden? Justify your answer.