Why do some words that appear to be feminine take a masculine article and vice-versa?
Although most words in Spanish are easily distinguished as masculine or feminine, there are many exceptions. 1. Words like, "tema", "programa", "sistema", and others terminate with the letter "a" but are all masculine nouns. Nouns that end in -ma, -ta, and -pa are often borrowed from Greek and therefore do not follow the expected pattern. Other notable exceptions include "mano", "foto", and "moto", all of which are feminine nouns. Some of these, like "foto" and "moto" are abbreviations of longer words that are clearly feminine (la fotografía, la motocicleta). Perhaps the most notable exception is "agua", which is in fact a feminine noun that takes a masculine article when used in singular form. "Agua" takes the masculine article when used in the singular to help with pronounciation (la agua when pronounced may sound like lagua). However when used as plural, it is correct to say las aguas. Also, when used with an adjective, the adjective should be feminine (i.e...el agua pura). As this short lesson shows, there are a few rule to help determine the gender of Spanish nouns. There are many other exceptions however that simply must be learned with practice.
A tall tree casts no shadow at 12:00pm. Assuming an angle of elevation of 10 degrees at 4:00pm and an approximately constant rate of increase in shadow length of 2 feet per minute, at what time does the length of the shadow equal the height of the tree? How tall is the tree?
First, we'll calculate the height of the tree. In reality shadow length won't increase at a constant rate, but to make the problem simpler, we'll make that assumption. Given a constant increase in length of 2 feet per minute over 240 minutes (there are 4 hours or 240 minutes between noon and 4:00pm) gives a 480 foot shadow at 4:00pm. Using the given information that the angle of elevation at 4:00pm is 10 degrees we can calculate the height of the tree as follows: tan(theta) = opposite/adjacent. In this case opposite = height of tree and adjacent = length of shadow. So, tan(10 degrees) = height/480. We then multiiply each side by 480 to solve for height. height = 480 * tan(10 degrees) In approximate form: height = 84.64 feet Next we'll find out at what time the length of the shadow equals the height of the tree. Since we know that the length of the shadow is initially zero feet, and the length increases at a rate of 2 feet per minute, we can simply divide the height of the tree by 2 to get the time elapsed from noon until the length of the shadow equals the height of the tree. 84.64/2 = 42.32 minutes = 42 minutes 19 seconds. Since the questions asks for the time of day when the length of the shadow equals the height of the tree, we'll just add 00:42:19 to 12:00pm to get a final answer of: 12:42:19.
Two common series are the Harmonic Series and the Alternating Harmonic Series. How can the end behavior of these series be determined? What tests can appropriately be applied to answer this question?
The Harmonic Series (given by the summation as n approaches infinity of 1 divided by n) is a Divergent Series. This can be shown with the Integral Test. The behavior of the Improper Integral evaluated from 1 to infinity of 1/x is divergent (the value approaches infinity). Interestingly, the Alternating Harmonic Series (given by the summation as n approaches infinity of (-1)^n divided by n) is a Convergent Series. This can be shown by the Alternating Series Test.