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Leno4ka [110]
3 years ago
9

The water in a pool evaporates at a rate of 16 gallons per week. What integer represents the change in the number of gallons of

water in the pool after 24 weeks
Mathematics
1 answer:
Sav [38]3 years ago
6 0
If 16 gallons of water evaporates per week and we have 24 weeks, we can jsut multiply it.
So 16*24=384 gallons of water evaporates.
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However, let's prove in a more formal way that

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First of all, it is injective: suppose x \neq y. Then, you trivially have \phi(x) \neq \phi(y), because they are two different matrices:

\phi(x) = \left[\begin{array}{cc}1&x\\0&1\end{array}\right],\quad \phi(y) = \left[\begin{array}{cc}1&y\\0&1\end{array}\right]

Secondly, it is trivially surjective: the matrix

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Finally, \phi and its inverse are both homomorphisms: if we consider the usual product between matrices to be the operation for the group G and the real numbers to be an additive group, we have

\phi (x+y) = \left[\begin{array}{cc}1&x+y\\0&1\end{array}\right] = \left[\begin{array}{cc}1&x\\0&1\end{array}\right] \cdot \left[\begin{array}{cc}1&y\\0&1\end{array}\right] = \phi(x) \cdot \phi(y)

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