Four thousand runners began the marathon. Eighty-nine percent of them completed the run. What percentage of the runner finished the marathon?

Step-by-step explanation:

Hi there!

Please see the answer in the picture.

Hope it helps!

1. Approach

One is given a trigonometric equation with and one is asked to prove that it is true. Using the attached image, combined with the knowledge of trigonometry, one can evaluate each trigonometric function. Then one can simplify each ratio to solve. To yield the most accurate result, one has to each of the ratios in a fractional form, rather than simplifying it into a decimal form. Remember the right angle trigonometric ratios, these ratios describe the relationship between the sides and angles in a right triangle. Such ratios are as follows,

[tex]sin(\theta)=\frac{opposite}{hypotenuse}\\\\cos(\theta)=\frac{adjacent}{hypotenuse}\\\\tan(\theta)=\frac{opposite}{adjacent}\\\\csc(\theta)=\frac{hypotenuse}{opposite}\\\\sec(\theta)=\frac{hypotenuse}{adjacent}\\\\cot(\theta)=\frac{adjacent}{opposite}[/tex]

Please note that the terms (opposite) and (adjacent) are relative to the angle uses in the ratio, however the term (hypotenuse) refers to the side opposite the right angle, this side never changes its name. Use these ratios to evaluate the trigonometric functions. Then simplify to prove the identity.

2. Problem (9)

[tex]\frac{sin(60)+cos(30)}{1+sin(30)+cos(60)}=sin(60)[/tex]

As per the attached image, the following statements regarding the value of each ratio can be made:

[tex]sin(60)=\frac{\sqrt{3}}{2}\\\\cos(30)=\frac{\sqrt{3}}{2}\\\\sin(30)=\frac{1}{2}\\\\cos(60)=\frac{1}{2}[/tex]

Substitute,

[tex]\frac{sin(60)+cos(30)}{1+sin(30)+cos(60)}=sin(60)[/tex]

[tex]\frac{\frac{\sqrt{3}}{2}+\frac{\sqrt{3}}{2}}{1+\frac{1}{2}+\frac{1}{2}}=\frac{\sqrt{3}}{2}[/tex]

Simplify,

[tex]\frac{\frac{\sqrt{3}}{2}+\frac{\sqrt{3}}{2}}{1+\frac{1}{2}+\frac{1}{2}}=\frac{\sqrt{3}}{2}[/tex]

[tex]\frac{\frac{2\sqrt{3}}{2}}{1+\frac{1}{2}+\frac{1}{2}}=\frac{\sqrt{3}}{2}[/tex]

[tex]\frac{\frac{2\sqrt{3}}{2}}{1+1}=\frac{\sqrt{3}}{2}[/tex]

[tex]\frac{\sqrt{3}}{1+1}=\frac{\sqrt{3}}{2}[/tex]

[tex]\frac{\sqrt{3}}{2}=\frac{\sqrt{3}}{2}[/tex]

Thus, this equation is true.

2. Problem (10)

Use a similar strategy to evaluate this equation,

[tex]\frac{1-cos(30)}{sin(30)}=\frac{1-cot(60)}{1+cot(60)}[/tex]

Use the attached image to evaluate the ratios.

[tex]cos(30)=\frac{\sqrt{3}}{2}\\\\sin(30)=\frac{1}{2}\\\\cot(60)=\frac{1}{\sqrt{3}}[/tex]

Substitute,

[tex]\frac{1-cos(30)}{sin(30)}=\frac{1-cot(60)}{1+cot(60)}[/tex]

[tex]\frac{1-\frac{\sqrt{3}}{2}}{\frac{1}{2}}=\frac{1-\frac{1}{\sqrt{3}}}{1+\frac{1}{\sqrt{3}}}[/tex]

Simplify,

[tex]\frac{1-\frac{\sqrt{3}}{2}}{\frac{1}{2}}=\frac{1-\frac{1}{\sqrt{3}}}{1+\frac{1}{\sqrt{3}}}[/tex]

[tex]\frac{\frac{2-\sqrt{3}}{2}}{\frac{1}{2}}=\frac{1-\frac{1}{\sqrt{3}}}{1+\frac{1}{\sqrt{3}}}[/tex]

[tex]\frac{\frac{2-\sqrt{3}}{2}}{\frac{1}{2}}=\frac{\frac{\sqrt{3}-1}{\sqrt{3}}}{1+\frac{1}{\sqrt{3}}}[/tex]

[tex]\frac{\frac{2-\sqrt{3}}{2}}{\frac{1}{2}}=\frac{\frac{\sqrt{3}-1}{\sqrt{3}}}{\frac{\sqrt{3}+1}{\sqrt{3}}}[/tex]

[tex]2-\sqrt{3}=\frac{\frac{\sqrt{3}-1}{\sqrt{3}}}{\frac{\sqrt{3}+1}{\sqrt{3}}}[/tex]

[tex]2-\sqrt{3}=\frac{\sqrt{3}-1}{\sqrt{3}}*\frac{\sqrt{3}}{\sqrt{3}+1}}[/tex]

[tex]2-\sqrt{3}=\frac{\sqrt{3}-1}{\sqrt{3}+1}[/tex]

Rationalize the denominator,

[tex]2-\sqrt{3}=\frac{\sqrt{3}-1}{\sqrt{3}+1}[/tex]

[tex]2-\sqrt{3}=\frac{\sqrt{3}-1}{\sqrt{3}+1}*\frac{\sqrt{3}-1}{\sqrt{3}-1}[/tex]

[tex]2-\sqrt{3}=\frac{(\sqrt{3}-1)^2}{3-1}[/tex]

[tex]2-\sqrt{3}=\frac{3-2\sqrt{3}+1}{2}[/tex]

[tex]2-\sqrt{3}=\frac{4-2\sqrt{3}}{2}[/tex]

[tex]2-\sqrt{3}=2-\sqrt{3}[/tex]

Therefore, this equation is also true.

9514 1404 393

Answer:

Step-by-step explanation:

Substitute the expressions.

After the first translation, the value of x is (x+n). Put that as the value of x in the second translation.

  x ⇒ x +s . . . . . . . . . the definition of the second translation

  (x+n) ⇒ (x+n) +s . . . the result after both translations

The same thing goes for y. After the first translation, its new value is (y+1).

  y ⇒ y +m . . . . . . . . the definition of the second translation

  (y+1) ⇒ (y+1) +m . . . the result of both translations

Then the composition of A followed by B is (x, y) ⇒ (x +n +s, y +1 +m).

__

The same reasoning applies. After the B translation, the x-coordinate is (x+s) and the y-coordinate is (y+m). Then the A translation changes these to ...

  x ⇒ x +n . . . . . . . . . . definition of translation A

  (x+s) ⇒ (x+s) +n . . . . translation A operating on point translated by B

  y ⇒ y +1 . . . . . . . . . . . definition of translation A

  (y+m) ⇒ (y+m) +1 . . . . translation A operating on point translated by B

The composition of B followed by A is (x, y) ⇒ (x + s + n, y + m + 1).

__

You should recognize that the sums (x+n+s) and (x+s+n) are identical. The commutative and associative properties of addition let us rearrange the order of the terms with no effect on the outcome.

The two translations give the same result in either order.

Answer:

-117 is irrational number

Answer:

Irrational

Step-by-step explanation:

Irrational number can't be written as a faction, -11pie can't be written as a fraction. Therefore it is a irrational number.

Answer:

BD = 10 √3

Step-by-step explanation:

ABC ∆ ,

BD ÷ 20 = sin 60

BD = 20 sin 60

BD = (20 √3) / 2

= 10 √3

Answer:

3rd graph

Step-by-step explanation:

Answer:

hope this help you

have a great day

Answer:

1≤f(x)≤5

Step-by-step explanation:

-1≤x≤1

-2≤2x≤2 (Multiplied by 2 both side)

-2+3≤2x+3≤2+3 (Adding three both sides)

1≤f(x)≤5

Answer:

In March she sent 752 texts

In April she sent 617

Step-by-step explanation:

The phone bill is 35 + additional texts, so in both bills the value relative to the texts are:

72.6 - 35 = 37.6 for March

65.85 - 35 = 30.85 for April

Each texts costs 0.05, so if we divide the value relative to the texts by 0.05 we will have the number of texts she sent:

37.6/0.05 = 752 texts

30.85/0.05 = 617 texts

Answer:

i use logical reasoning

Answer:

33/14

Step-by-step explanation:

[tex] - 10 \frac{2}{7} + ( - 4 \frac{4}{11} )[/tex]

[tex] = - \frac{72}{7} \div - ( \frac{48}{11} )[/tex]

[tex] = \frac{72}{7} \times \frac{11}{48} [/tex]

[tex] = \frac{3}{7} \times \frac{11}{2} [/tex]

[tex] = \frac{33}{14} [/tex]

stan dreamcatcher

Answer:

4

Step-by-step explanation:

Let's simplify step-by-step.

−2z2+4z+2z2

Combine Like Terms:

=−2z2+4z+2z2

=(−2z2+2z2)+(4z)

=4z

Answer:

4z

Hope this helps <3 Need thanks?

Comment /hearthelp

[tex]\\ \sf\longmapsto -2z^2+4z+2z^2[/tex]

[tex]\\ \sf\longmapsto -2z^2+2z^2+4z[/tex]

[tex]\\ \sf\longmapsto (-2+2)z^2+4z[/tex]

[tex]\\ \sf\longmapsto 0z^2+4z[/tex]

[tex]\\ \sf\longmapsto 4z[/tex]

the answer is

A. 28, 25, 22

Answer:

The area of the circle is 100 square units.

Step-by-step explanation:

We are given that the circumference of a circle is 20π, and we want to determine its area.

Recall that the circumference of a circle is given by the formula:

[tex]\displaystyle C = 2\pi r[/tex]

Substitute:

[tex]20 \pi = 2 \pi r[/tex]

Solve for the radius:

[tex]\displaystyle r = \frac{20\pi}{2\pi} = 10[/tex]

The area of a circle is given by:

[tex]\displaystyle A = \pi r^2[/tex]

Since the radius is 10 units:

[tex]\displaystyle A = \pi (10)^2[/tex]

Evaluate:

[tex]\displaystyle A = 100\pi\text{ units}^2[/tex]

In conclusion, the area of the circle is 100 square units.

Answer:

6.8 gallions i believe. im not quite sure

Answer:

Solution given:

x^5=225

we have

x=[tex] \sqrt[5]{225} [/tex]

x=2.9541

Hello!

[tex] \bf {x}^{5} = 225[/tex]

[tex] \bf x = \sqrt[5]{225} [/tex]

[tex] \bf x ≈2.95418[/tex]

Answer: x ≈ 2,95418

Good luck! :)

Answer:

p = -5

every element in the sequence is created by subtracting 3 from the pervious element.

Step-by-step explanation:

I am not sure you put every necessary information here.

but I'd the visible information is truly everything, then it's is trivial.

the difference between 4 and 1 is ... well, -3. meaning we subtract 3 from 4 to get 1.

we suspect this is the rule and keep trying.

1 -3 = -2

hey it works.

and -8 -3 = -11

hey, also correct.

and the difference between -2 and -8 is -6, and when we place another item in between (p), we cut that in half again to -3. so, it is all consistent.

therefore,

p = -2 -3 = -5

the rule is

an = an-1 - 3

Answer:

A

Step-by-step explanation:

pgt states if the square of the length of the longest side of a triangle is equal to the sum of the squares of the other two sides, then the triangle is a right triangle.

Answer:

Does the answer help you?

Answer:

A " (1,-2)

B " (4,0)

C " (6,-3)

Step-by-step explanation:

Hope it helped.

° ° °

Answer:

x=61.71428 or 61 5/7

Step-by-step explanation:

We can use a ratio to solve

7 pieces              12 pieces

-----------------   = ---------------

36 seconds          x seconds

Using cross products

7x = 36*12

7x = 432

Divide by 7

7x/7 = 432/7

61 5/7

x=61.71428

Answer:

66

Step-by-step explanation:

Answer:

O D. 3x2 + 2x-3

Answer:

A.7x2-5x questions 2of 20

Step-by-step explanationIf you are reading this say

thank u