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Ilya [14]
2 years ago
7

If sin(2x) = cos (x + 30%), what is the value of x?!

Mathematics
1 answer:
fiasKO [112]2 years ago
8 0

Answer:

  x ∈ {20°, 120°, 140°, 260°}

Step-by-step explanation:

I find a graphing calculator useful for solving equations of this sort. Subtracting cos(x+30°) from both sides gives an expression that is zero when x has the appropriate value. The calculator is very good at finding zeros.

__

We can make use of the formula for the sum of two sines.

  sin(a) +sin(b) = 2sin((a+b)/2)cos((a-b)/2)

Then we can rewrite ...

  sin(2x) -sin(x +120°) = 0

as ...

  2sin((2x-(x+120°))/2)·cos((2x+(x+120°))/2) = 0

  sin((x -120°)/2)·cos((3x +120°)/2) = 0 . . . . . divide by 2 and simplify a bit

This resolves to two equations according to the zero product rule.

  x/2 -60° = k·180° . . . . . for some integer k

  x = 120° +k·360° . . . . . for some integer k

And the other equation is ...

  3x/2 +60° = (2k +1)·90° . . . . . for some integer k

  3x/2 = k·180° +30° . . . . . . . . subtract 60°, eliminate parentheses

  3x = k·360° +60°

  x = k·120° +20° . . . . . for some integer k

The values shown in the Answer section above are the values of x in the range 0 to 360°: 20°, 120°, 140°, 260°.

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A section of wall is being framed. A model of the framing work is shown below. Vertical and parallel lines c, d, and e are cut b
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Answer:

A section of wall is being framed. A model of the framing work is shown below. Vertical and parallel lines c, d, and e are cut by diagonal transversal b. The uppercase right angle formed by the intersection of lines b and c is angle A. The uppercase left angle formed by the intersection of lines d and b is 125 degrees. Which best describes the relationship between the 125° angle and angle A? They are same side interior angles. Angle A measures 55°. They are alternate interior angles. Angle A measures 125°. They are vertical angles. Angle A measures 125°. They are corresponding angles. Angle A measures 55°.

angle D

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3 years ago
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When functions are defined by more than one​ equation, they are called?
BARSIC [14]
When functions are defined by more then one equation, they are called piecewise defined functions
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3 years ago
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Suppose that the world's current oil reserves is R = 1820 billion barrels. If, on average, the total reserves
iVinArrow [24]

Answer:

A. R = -18t + 1,820

B. 1,568 billions of barrels

C. approximately 101.11 years

Step-by-step explanation:

To make our equation, we'll use the form R = mt + b. M represents how many billion barrels of oil are being lost each year, which we know is 18 billion. So -18 will be our m. B is how many total barrels of oil there are, which is 1,820. So 1,820 will be our b. Now the equation looks like this:

R = -18t + 1,820

We can use this equation to answer Part B.

Replace the t with 14:

R = -18(14) + 1,820

Now solve for R:

R = -18(14) + 1,820

R = -252 + 1,820

R = 1,568

14 years from now, there will be 1,568 billions of barrels left.

To solve part C, we need to find how many years it will take for all of the oil to be used up. After it's all used up, the total amount of oil will be 0, so we can replace R with 0 and then solve for t:

0 = -18t + 1,820

Subtract 1,820 from both sides to isolate -18t:

0 - 1,820 = -18t + 1,820 - 1,820

-1,820 = -18t

Divide both sides by -18 to isolate the t:

-1,820/-18 = -18t/-18

101.11 = t

After approximately 101.11 years, all of the oil will be used up.

3 0
2 years ago
Andrew used a pedometer to measure the
pychu [463]

Answer:

  13.25 mi

Step-by-step explanation:

The largest number on the list is 13.25. The greatest distance was 13.25 miles.

_____

In order from smallest to largest, the distances are ...

  4.9, 5.95, 8.9, 9.8, 13.25

8 0
3 years ago
the half-life of chromium-51 is 38 days. If the sample contained 510 grams. How much would remain after 1 year?​
madam [21]

Answer:

About 0.6548 grams will be remaining.  

Step-by-step explanation:

We can write an exponential function to model the situation. The standard exponential function is:

f(t)=a(r)^t

The original sample contained 510 grams. So, a = 510.

Each half-life, the amount decreases by half. So, r = 1/2.

For t, since one half-life occurs every 38 days, we can substitute t/38 for t, where t is the time in days.

Therefore, our function is:

\displaystyle f(t)=510\Big(\frac{1}{2}\Big)^{t/38}

One year has 365 days.

Therefore, the amount remaining after one year will be:

\displaystyle f(365)=510\Big(\frac{1}{2}\Big)^{365/38}\approx0.6548

About 0.6548 grams will be remaining.  

Alternatively, we can use the standard exponential growth/decay function modeled by:

f(t)=Ce^{kt}

The starting sample is 510. So, C = 510.

After one half-life (38 days), the remaining amount will be 255. Therefore:

255=510e^{38k}

Solving for k:

\displaystyle \frac{1}{2}=e^{38k}\Rightarrow k=\frac{1}{38}\ln\Big(\frac{1}{2}\Big)

Thus, our function is:

f(t)=510e^{t\ln(.5)/38}

Then after one year or 365 days, the amount remaining will be about:

f(365)=510e^{365\ln(.5)/38}\approx 0.6548

5 0
2 years ago
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