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aivan3 [116]
3 years ago
10

one-third of a number x is equal to 22 less than the number. Write and solve an equation to find the number.

Mathematics
1 answer:
vitfil [10]3 years ago
4 0
1/3x=x-22
Add 22 to both sides to get 22+1/3x=x
Subtract 1/3x from both sides to get 22=2/3x
Multiply each side by 3/2 to get x=33!
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Question (c)! How do I know that t^5-10t^3+5t=0?<br> Thanks!
astra-53 [7]
(a) By DeMoivre's theorem, we have

(\cos\theta+i\sin\theta)^5=\cos5\theta+i\sin5\theta

On the LHS, expanding yields

\cos^5\theta+5i\cos^4\theta\sin\theta-10\cos^3\theta\sin^2\theta-10i\cos^2\theta\sin^3\theta+5\cos\theta\sin^4\theta+i\sin^4\theta

Matching up real and imaginary parts, we have for (i) and (ii),


\cos5\theta=\cos^5\theta-10\cos^3\theta\sin^2\theta+5\cos\theta\sin^4\theta
\sin5\theta=5\cos^4\theta\sin\theta-10\cos^2\theta\sin^3\theta+\sin^5\theta

(b) By the definition of the tangent function,

\tan5\theta=\dfrac{\sin5\theta}{\cos5\theta}
=\dfrac{5\cos^4\theta\sin\theta-10\cos^2\theta\sin^3\theta+\sin^5\theta}{\cos^5\theta-10\cos^3\theta\sin^2\theta+5\cos\theta\sin^4\theta}

=\dfrac{5\tan\theta-10\tan^3\theta+\tan^5\theta}{1-10\tan^2\theta+5\tan^4\theta}
=\dfrac{t^5-10t^3+5t}{5t^4-10t^2+1}


(c) Setting \theta=\dfrac\pi5, we have t=\tan\dfrac\pi5 and \tan5\left(\dfrac\pi5\right)=\tan\pi=0. So

0=\dfrac{t^5-10t^3+5t}{5t^4-10t^2+1}

At the given value of t, the denominator is a non-zero number, so only the numerator can contribute to this reducing to 0.


0=t^5-10t^3+5t\implies0=t^4-10t^2+5

Remember, this is saying that

0=\tan^4\dfrac\pi5-10\tan^2\dfrac\pi5+5

If we replace \tan^2\dfrac\pi5 with a variable x, then the above means \tan^2\dfrac\pi5 is a root to the quadratic equation,

x^2-10x+5=0

Also, if \theta=\dfrac{2\pi}5, then t=\tan\dfrac{2\pi}5 and \tan5\left(\dfrac{2\pi}5\right)=\tan2\pi=0. So by a similar argument as above, we deduce that \tan^2\dfrac{2\pi}5 is also a root to the quadratic equation above.

(d) We know both roots to the quadratic above. The fundamental theorem of algebra lets us write

x^2-10x+5=\left(x-\tan^2\dfrac\pi5\right)\left(x-\tan^2\dfrac{2\pi}5\right)

Expand the RHS and match up terms of the same power. In particular, the constant terms satisfy

5=\tan^2\dfrac\pi5\tan^2\dfrac{2\pi}5\implies\tan\dfrac\pi5\tan\dfrac{2\pi}5=\pm\sqrt5

But \tanx>0 for all 0, as is the case for x=\dfrac\pi5 and x=\dfrac{2\pi}5, so we choose the positive root.
3 0
3 years ago
James has to fill 40 water bottles for the team. Each bottle holds 500 millimeters of water. How many liters of water does James
Advocard [28]

Answer:

20 liters of water

he will need 20 liters of water

Step-by-step explanation:

if james has 40 bottles that hold 500 milliliter then he needs 20,000 milliliters of water or 20 liters. :)

6 0
3 years ago
Read 2 more answers
PLEASE HELP ME ASAP!! I had trouble with this and it is now over due
erica [24]

Answer:

Shira: x = 8

Samuel: m = 0

Step-by-step explanation:

Shira's mistake was that she subtracted 2 from both sides instead of adding to on both sides.

Correct Solving:

2x - 2 = 14

Add 2 to both sides;

2x = 16

Divide both sides by 2;

x = 8

Samuel's mistake was that when he distributed -2 to 8m and 8 he put the wrong sign for -2 * 8.

Correct Solving:

-2(8m + 8) = -16

Distribute;

-16m - 16 = -16

Add 16 to both sides;

-16m = 0

Divide both sides by -16;

m = 0

3 0
3 years ago
Which of the following have at least two congruent parallel bases? all that apply. A. Cylinder B. Circle C. Cone D. Cube E. None
Nookie1986 [14]

Answer:

Option A. Cylinder.

Option D. Cube

Step-by-step explanation:

In this question we have to tell that which are the figures have parallel bases.

Cylinder: Cylinder has two parallel congruent circular bases at the top and at the bottom.

Circle : It's a two dimensional structure and doesn't has any top or bottom.

Cone : It has only one circular bottom.

Pyramid : Pyramids have only one base circular or rectangular.

Cube : Cube is a three dimensional figure having minimum two congruent sides parallel to each other.

Therefore Option A. Cylinder  and Option D. Cube are the answers.

7 0
3 years ago
Read 2 more answers
Find the ordered pair whose x-coordinate is 5. 4x=6y+8
sukhopar [10]

(6, 2 )

substitute x = 5 into the equation and solve for y

6y + 8 = 20 ( subtract 8 from both sides )

6y = 12 ( divide both sides by 6 )

y = \frac{12}{6} = 2

the ordered pair is (5, 2 )


6 0
3 years ago
Read 2 more answers
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