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In an isosceles triangle, the longest side measures c units. The other sides of the triangle measure a units. Which equation can be used to determine if the triangle is a right triangle?
a squared = c squared
2 a squared = c squared
a Superscript 4 Baseline = c squared
2 a = c

Asmall (but heavy) particle placed in a glass of water will follow a zigzag motion because the particle will bounce off of the water molecules it meets. this is called brownian motion. a physicist simulates this on a computer, by varying the distance a particle can travel (called the mean free length), on average, before it collides with a water molecule and assigning the change in motion to be one of 8 directions, each with a similar probability. by running the simulated particle (with the same mean free length) many times she determines that it should take 15 seconds, on average, for the particle to fall to the bottom, with a standard deviation of 1.5 seconds. next she lets a real particle fall through a glass of water and finds that it took 18 seconds. what does she conclude, and why?

The biologist has a second culture to examine. she knows that the population of the culture doubles every 15 minutes. after 1 hour and 15 minutes, her assistant found that 80,000 bacteria were present. a) what was the size of the initial population? b) predict the size of the culture at t=3 hours. what was the size of the population at 40 minutes? explain and justify your answers. c) create a graph of the population as a function of time. find an equation that can be used to predict the size of the population at any time t. d) examine the rate at which the bacteria culture is growing. how fast is the culture growing after 1 hour? after 1.5 hours? after 2 hours? use a time interval of h = 0.01 hours to estimate these rates. interpret these rates in terms of the context of the problem situation. how do these three rates compare?

Un used xx milliliters, or 90\%90%, of his favorite bottle of hot sauce. which of the following expressions could represent the volume of a full bottle of jun's favorite hot sauce, in milliliters? choose 2 answers: choose 2 answers: (choice a) a \dfrac{x}{0.9} 0.9 x (choice b) b \dfrac{10}{9}x 9 10 x (choice c) c 0.9x0.9x (choice d) d xx (choice e) e x+0.1x+0.1