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Firlakuza [10]
3 years ago
5

Stephon has a square brick patio. He wants to reduce the width by 4 feet and increase the length by 4 feet.

Mathematics
1 answer:
alekssr [168]3 years ago
4 0

Answer:  The expressions for the length and width of the new patio are

\ell=x+4,~~w=x-4.

And the area of the new patio is 384 sq. feet.

Step-by-step explanation:  Given that Stephen has a square brick patio. He wants to reduce the width by 4 feet and increase the length by 4 feet.

The length of one side of the square patio is represented by x.

We are to write the expressions for the length and width of the new patio and then to find the area of the new patio if the original patio measures 20 feet by 20 feet.

Since Stephen wants to reduce width of the patio by 4 feet, so the width of the new patio will be

w=(x-4)~\textup{feet}.

The length of the patio is increased by 4 feet, so the length of the new patio will be

\ell=(x+4)~\textup{feet}.

Now, if the original patio measures 20 feet by 20 feet, then we must have

w=x-4=20-4=16~\textup{feet}

and

\ell=x+4=20+4=24~\textup{feet}.

Therefore, the area of the new patio is given by

A_n=\ell \times w=24\times16=384~\textup{sq. feet}.

Thus, the expressions for the length and width of the new patio are

\ell=x+4,~~w=x-4.

And the area of the new patio is 384 sq. feet.

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The first ratio I have can be found using \frac{\pi}{6} in the first rotation of the unit circle.

The second ratio I have can be found using \frac{7\pi}{6} you can see this is on the same line as the \frac{\pi}{6} so you could write \frac{7\pi}{6} as \frac{\pi}{6}+\pi.

So this means the following:

\tan(x-\frac{\pi}{8})=\frac{1}{\sqrt{3}}

is true when x-\frac{\pi}{8}=\frac{\pi}{6}+n \pi

where n is integer.

Integers are the set containing {..,-3,-2,-1,0,1,2,3,...}.

So now we have a linear equation to solve:

x-\frac{\pi}{8}=\frac{\pi}{6}+n \pi

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x=\frac{\pi}{6}+\frac{\pi}{8}+n \pi

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Now I just notice that it said find all the solutions in the interval [0,2\pi).

So if \sqrt{3} \tan(x-\frac{\pi}{8})-1=0 and we let u=x-\frac{\pi}{8}, then solving for x gives us:

u+\frac{\pi}{8}=x ( I just added \frac{\pi}{8} on both sides.)

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