Find all solutions and solutions in interval[0,2pi)

Mathematics

1 answer:

frutty [35]3 years ago

7 0

$\bf sin(2\theta)=2sin(\theta)cos(\theta)\\\\
-------------------------------\\\\
sin(2x)-\sqrt{3}cos(x)=0\implies 2sin(x)cos(x)-cos(x)\sqrt{3}=0
\\\\\\
\stackrel{\stackrel{common}{factor}}{cos(x)}[2sin(x)-\sqrt{3}]=0\\\\
-------------------------------\\\\$

$\bf cos(x)=0\implies \measuredangle x=
\begin{cases}
\frac{\pi }{2}\\\\
\frac{3\pi }{2}
\end{cases}\\\\
-------------------------------\\\\
2sin(x)-\sqrt{3}=0\implies 2sin(x)=\sqrt{3}\implies sin(x)=\cfrac{\sqrt{3}}{2}
\\\\\\
\measuredangle x=
\begin{cases}
\frac{\pi }{3}\\\\ \frac{2\pi }{3}
\end{cases}$

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Theresa Morgan a chemist has a 10% hydrochloric acid solution and a 65% hydrochloric acid solution. How many liters of each shou

prisoha [69]

<u>ANSWER: </u>

170 liters of 10% hydrochloric acid must be mixed with 1700 liters of 65% hydrochloric acid solution to obtain 1870 liters of 60% solution.

<u>SOLUTION: </u>

First, set up a table. Fill in the unknowns with variables x and y. The table is attached below

From this, we can easily set up the two equations.

Sum of values of two acids = Value of mixture

0.1x + 0.65y = 1122

For convenience, we'll multiply the entire equation by 100,

10x + 65y = 112200 ------ (1)

Now, Sum of amounts of each acid = Amount of mixture

x + y = 1870 ---------(2)

multiply (2) with 10 for easy calculation and derive the equation into one variable.

10x + 10y = 18700

Subtracting equation (2) from (1),

( + ) 10x + 65y = 112200

( − ) 10x + 10y = 18700

we get the solution as,

0 + 55y = 93500

Thus, 55y = 93500

y = 1700

Substituting y = 1700 in (2),

x + 1700 = 1870

x = 1870 - 1700

x = 170

So, we have x = 170 and y = 1700

We can conclude that 170 liters of 10% hydrochloric acid must be mixed with 1700 liters of 65% hydrochloric acid solution to obtain 1870 liters of 60% solution.