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3 years ago

The wolfpack has one 11 games and lost 9 games this season. what percent of its games did the wolfpack win?

1 answer:

4 0

In fractions they had won 11/20 games. Then if you convert the fraction to a percent you would get 55% after multipling by 5/5 to get the denominator of 100.

The answer would be 55%.

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I need help with my math homework. The questions is: Find all solutions of the equation in the interval [0,2π).

Aleksandr-060686 [28]

Answer:

$\frac{7\pi}{24}$ and $\frac{31\pi}{24}$

Step-by-step explanation:

$\sqrt{3} \tan(x-\frac{\pi}{8})-1=0$

Let's first isolate the trig function.

Add 1 one on both sides:

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

Divide both sides by $\sqrt{3}$ :

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

Now recall $\tan(u)=\frac{\sin(u)}{\cos(u)}$ .

$\frac{1}{\sqrt{3}}=\frac{\frac{1}{2}}{\frac{\sqrt{3}}{2}}$

$\frac{1}{\sqrt{3}}=\frac{-\frac{1}{2}}{-\frac{\sqrt{3}}{2}}$

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:

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

Add $\frac{\pi}{8}$ on both sides:

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

Find common denominator between the first two terms on the right.

That is 24.

$x=\frac{4\pi}{24}+\frac{3\pi}{24}+n \pi$

$x=\frac{7\pi}{24}+n \pi$ (So this is for all the solutions.)

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.)

So recall $0\le x$ .

Then $0 \le u+\frac{\pi}{8}$ .

Subtract $\frac{\pi}{8}$ on both sides:

$-\frac{\pi}{8}\le u$

Simplify:

$-\frac{\pi}{8}\le u$

So we want to find solutions to:

$\tan(u)=\frac{1}{\sqrt{3}}$ with the condition:

$-\frac{\pi}{8}\le u$

That's just at $\frac{\pi}{6}$ and $\frac{7\pi}{6}$

So now adding $\frac{\pi}{8}$ to both gives us the solutions to:

$\tan(x-\frac{\pi}{8})=\frac{1}{\sqrt{3}}$ in the interval:

$0\le x$ .

The solutions we are looking for are:

$\frac{\pi}{6}+\frac{\pi}{8}$ and $\frac{7\pi}{6}+\frac{\pi}{8}$

Let's simplifying:

$(\frac{1}{6}+\frac{1}{8})\pi$ and $(\frac{7}{6}+\frac{1}{8})\pi$

$\frac{7}{24}\pi$ and $\frac{31}{24}\pi$

$\frac{7\pi}{24}$ and $\frac{31\pi}{24}$

5 0

3 years ago

What is the Area of the circle?

r-ruslan [8.4K]

The answer is 254.35 in square

4 0

3 years ago

Read 2 more answers

Which real-world scenario can be described by the algebraic expression 4w? withdrawing w dollars from the bank and giving 4 doll

ANTONII [103]

Answer:

It's B

Step-by-step explanation:

3 0

3 years ago

Read 2 more answers

Solve x2 – 10x + 25 = 35 for x.

Galina-37 [17]

$x^2-10x+25=35\\\\x^2-2\cdot5\cdot x + 5^2=35\ \ \ |\text{use:}\ (a-b)^2=a^2-2ab+b^2\\\\(x-5)^2=35\to x-5=\pm\sqrt{35}\ \ \ |+5\\\\\boxed{x=5-\sqrt{35}\ \vee\ x=5+\sqrt{35}}$

$\text{other method}\\\\x^2-10x+25=35\\\\x^2-5x-5x+25=35\\\\x(x-5)-5(x-5)=35\\\\(x-5)(x-5)=35\\\\(x-5)^2=35\\.....$

3 0

4 years ago

A graph has number of days on the x-axis and number of canned goods on the y-axis. A line goes through points (2, 699) and (8, 3

Kitty [74]

Answer:

Slope:-63

Y-intercept: (0,825)

Step-by-step explanation:

Slope= y2-y1 ÷ x2-x1

321-699 ÷ 8-2

-378 ÷ 6 =-63

Y-intercept:

y=-63x+b

to get b (y-intercept) plug in one of the points and solve.

699=-63(2)+b

699=-126+b

699+126=-126+126+b

699+126=0+b

825=b

5 0

3 years ago

Read 2 more answers