2017-09-11 Basic Practice Problems Online

The area of the large square is 100 and all of the small, blue squares are congruent.

What is the total area shaded blue?

Which is larger, $\sqrt{ 5 + \frac5{24} }\quad \text{ or }\quad 5 \times \sqrt{\frac5{24} }\,?$

A wrecking ball is a heavy metal mass that's swung, from a crane, into structures to demolish them. Suppose we release a ball of mass $M$ from initial angle $\theta$ and measure the speed at the lowest point in its swing to be $v_\textrm{max}.$

If we replace the wrecking ball with one of mass $2M,$ keeping the length of the chain and the angle $\theta$ at which the ball is released constant, how will $v_\textrm{max}$ be affected?

Details and Assumptions:

Neglect air resistance.
The mass of the chain is zero.

Two players are playing a shortened version of badminton: a 6-point, 3-game match with no deuce. Specifically, in each game, the player who first scores 6 points wins. The winner of the match is the player who first wins 2 out of 3 games.

Is it possible for the loser to have accumulated more points than the winner?

$\large 1 \; \square \; \dfrac{1}{2} \; \square \; \dfrac{1}{3}\; \square \; \dfrac{1}{4}\; \square \; \dfrac{1}{5} \; \square \; \dfrac{1}{6} \; \square \; \dfrac{1}{7} \; \square \; \dfrac{1}{8} \; \square \; \dfrac{1}{9} \; \square \; \dfrac{1}{10} \; \square \; \dfrac{1}{11} \; \square \; \dfrac{1}{12}= 0$

Can you fill the boxes with $+ \text{ and } -$ to make this equation true?

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