Suppose x is any positive number. Circle 1 has a center at (−2, 7) and a radius of 8x. Circle 2 has a center at (−3, 0) and a radius of 5x. Why is Circle 1 similar to Circle 2?Circle 1 and Circle 2 have the same area, and Circle 1 has a radius 0.625 times longer than Circle 2.Circle 1 is a translation of 1 unit right and 7 units up from Circle 2, and a dilation of Circle 2 with a scale factor of 1.6.Circle 1 is a translation of 1 unit right and 7 units up from Circle 2, and a dilation of Circle 2 with a scale factor of 0.625.Circle 1 and Circle 2 have the same circumference, and Circle 1 has a radius 1.6 times the length of Circle 2's radius.

➷ The formula for a right triangle is:

(b x h) / 2

We know this:

(12 x n) / 2 = 30

Multiply both sides by 2:

12 x n = 60

Divide both sides by 12:

n = 5

Now we know the height is 5

We can now use Pythagoras' theorem to calculate the hypotenuse:

c^2 = 5^2 + 12^2

c^2 = 169

Square root it:

c = 13

The hypotenuse is 13 ft

➶ Hope This Helps You!

➶ Good Luck (:

➶ Have A Great Day ^-^

↬ ʜᴀɴɴᴀʜ ♡

Answer:

The formula for working out the area of a triangle:

(b x h) / 2

[tex](12*h)/2=30\\(*2)\\12*h=60h\\h=5[/tex]

Pythagoras' theorem to calculate the hypotenuse:

c² = 5² + 12²

c² = 169

c = 13

The hypotenuse is 13 ft

Step-by-step explanation:

Answer:

The volume of a rectangular prism is [tex]28\frac{7}{8}\ cm^{3}[/tex]

Step-by-step explanation:

we know that

The volume of the rectangular prism is equal to

[tex]V=BHL[/tex]

Convert the given dimensions to an improper fractions

[tex]B=3\frac{1}{2}\ cm=\frac{3*2+1}{2}=\frac{7}{2}\ cm[/tex]

[tex]H=1\frac{1}{2}\ cm=\frac{1*2+1}{2}=\frac{3}{2}\ cm[/tex]

[tex]L=5\frac{1}{2}\ cm=\frac{5*2+1}{2}=\frac{11}{2}\ cm[/tex]

substitute in the formula

[tex]V=(\frac{7}{2})(\frac{3}{2})(\frac{11}{2})=\frac{231}{8}\ cm^{3}[/tex]

Convert to mixed number

[tex]\frac{231}{8}=\frac{224}{8}+\frac{7}{8}=28\frac{7}{8}\ cm^{3}[/tex]

Answer:

Step-by-step explanation:

Your car breaks down unexpectedly and you must have it repaired immediately.  That's an example of a situation where you might use money from a financial reserve.

Answer:

B - closing costs are increased

Step-by-step explanation:

All of the other options are advantages to purchasing points. You want lower monthly payments (a), a tax break (c), and lower interest rate (d)

Answer:                

The correct answer is option B.closing costs are increased.

Step-by-step explanation:                        

Mortgage points or discount points refers to the fees paid directly to the lender during closing in exchange for a reduced interest rate.

In this way the buyer buys down the rate, that helps in reducing his monthly mortgage payments.

So, option A, C and D are the advantages of purchasing discount points.

Answer:

Step-by-step explanation:

The outcome of the game is

Toss 1               Toss 2      Result

H                         H               18

H                         T                 -1

T                         H                -1

T                         T                -20

His only winning position is H ---- H

That means he has only a 1/4 chance in winning. He shouldn't play at all. It might be a fair coin, but it is not a fair game.

Answer:

The answer is 9.6

hope this helps!

Answer:

9.9 cm

Step-by-step explanation:

We can use the Pythagorean theorem to find the length of CY

a^2 + b^2 = c^2

CY^2 + YZ^2 = CZ^2

YZ = XY  since CZ is the perpendicular bisector

YZ = 5

CZ = 7

CY ^2 + 5^2 = 7^2

CY^2 +25 = 49

Subtract 25 from each side

CY^2 = 49-25

CY^2 = 24

Take the square root of each side

sqrt(CY^2) = sqrt(24)

CY = 4.898979

CY = 4.9 cm

We want the length of WY

WY = WC + CY

WC is a radius which is 5 cm

WY = 5cm + 4.9cm

WY = 9.9 cm

Answer:

We should work backwards we need to find YC+CW to get YW

angle bisector theorem means that ZC and XC are equal

then we can use the Pythagorean theorem to get YC

5^2 + x = 7^2

YC= √13

CW = 7 because they are both the radius of a circle

YW= 7+√13

YW=10.60 (rounded)

Answer:

a. 61.41%

b. 27 minutes

Step-by-step explanation:

a:  Find the z-score for the situation.  

µ = 19

x-bar = 20

σ = 3.5

  z = (20 - 19)/3.5 = 0.29

The p-value for z = 0.29 is 0.6141, so 61.41% of people will get this discount

b:  They want no more than 2% to get the discount, so they want less than 98% getting the discount.  The z-score for 98% (0.98 as a decimal) is 2.055

*You need to look at the chart and find where 0.98 would be.  It's between a z-score of 2.05 and 2.06.  

The z-score is    2.055 = (x - 19)/3.5      We are solving for the time for this one.  So solve for x...

              7.1925 = x - 19      (multiply both sides by 3.5)

             26.1925 = x    (add 19 to both sides)

So 26.1925 minutes, or about 26 minutes, 12 seconds, so round up to 27 minutes because they want less than 2%.   I chose 27 minutes because no places give odd wait times like 26 minutes 12 seconds.

A) The percent of customers receive the service for​ half-price is; 61.41%

B) The time to make the guaranteed limit is; 26 minutes 12 seconds

A) We are given;

Population mean; µ = 19

Sample mean; x' = 20

Standard deviation; σ = 3.5

Thus, z-score is;

z = (20 - 19)/3.5

z = 0.29

From online p-value from s-score calculator,  the p-value for z = 0.29 is;

p = 0.6141 = 61.41%

B) We are told that they now want more than 2% to get the discount. This means that they want less than 98% or 0.98 getting the discount.  

The z-score for 0.98 is; z = 2.055

Thus, using z-score formula, we have;

2.055 = (x' - 19)/3.5    

x' - 19 = 3.5 * 2.055

x' - 19 = 7.1925

x' = 26.1925 = 26 minutes 12 seconds

Read more about P-value at; https://brainly.com/question/4621112

Answer:

x = log base 4 of 2

Step-by-step explanation:

I'm assuming this is truly an exponential function as you state and not a mistype.

6^x = 2

Take the log of each side:

x · log(6) = log(2)

Divide each side by log(6):

x = log(2) / log(6)

x = 0.30103 / 0.77815

x = 0.3869... (rounded)

Answer:

The answer is circle; (x')² + (y')² - 4 = 0

Step-by-step explanation:

* At first lets talk about the general form of the conic equation

- Ax² + Bxy + Cy²  + Dx + Ey + F = 0

∵ B² - 4AC < 0 , if a conic exists, it will be either a circle or an ellipse.

∵ B² - 4AC = 0 , if a conic exists, it will be a parabola.

∵ B² - 4AC > 0 , if a conic exists, it will be a hyperbola.

* Now we will study our equation:

* 2x² + 2y² = 8

∵ A = 2 , B = 0 , C = 2

∴ B² - 4AC = (0) - 4(2)(2) = -16 < 0

∵ B² - 4AC < 0

∴  it will be either a circle or an ellipse

* Lets use this note to chose the correct figure

- If A and C are equal and nonzero and have the same sign,

 then the graph is a circle.

- If A and C are nonzero, have the same sign, and are not equal

 to each other, then the graph is an ellipse.

∵ A = 2 and C = 2

∴ The graph is a circle.

∵ D and E = 0

∴ The center of the circle is the origin (0 , 0)

∵ Ф = 30°

∴ The point (x , y) will be (x' , y')

- Where x = x'cosФ - y' sinФ and y = x'sinФ + y'cosФ

∴ x = x'cos(30°) - y'sin(30°)

∴ y = x'sin(30°) + y'cos(30°)

∴ x = (√3/2)x' - (1/2)y' and y = (1/2)x' + (√3/2)y'

∴ [tex]x=\frac{\sqrt{3}x'-y'}{2}[/tex]

∴ [tex]y=\frac{x'+\sqrt{3}y'}{2}[/tex]

* Lets substitute x and y in the first equation

∴ [tex]2(\frac{\sqrt{3}x'-y'}{2})^{2}+2(\frac{x'+\sqrt{3}y'}{2})^{2}=8[/tex]

* Use the foil method

∴ [tex]2(\frac{3x'^{2}-2\sqrt{3}x'y'+y'^{2}}{4})+2(\frac{x'^{2}+2\sqrt{3}x'y'+3y'^{2}}{4})=8[/tex]

* Open the brackets

∴ [tex]\frac{3x'^{2}-2\sqrt{3}x'y'+y'^{2}+x'^{2}+2\sqrt{3}x'y'+3y'^{2}}{2}=8[/tex]

* Collect the like terms

∴ [tex]\frac{4x'^{2}+4y'^{2}}{2}=8[/tex]

* Simplify the fraction

∴ 2(x')² + 2(y')²= 8

* Divide each side by 2

∴ (x')² + (y')² = 4

∴ The equation of the circle is (x')² + (y')² = 4

* The general equation of the circle is (x')² + (y')² - 4 = 0  

 after rotation 30° about the origin

* Look to the graph

- The blue circle for the equation 2x² + 2y² = 8

- The blue circle for equation (x')² + (y')² - 4 = 0

* That is because the two circles have same centers and radii

- The green line is x' and the purple line is y'

Answer:

The answer is D

Good luck on the Ed-genuity test

Answer:

The answer is 1/2 which i think that's what you meant from C.

Step-by-step explanation:

A standard deck of cards has 52 cards. 13 of these cards are spades and 13 are diamonds.

the answer to your question is c :]

Answer:

b. [tex]\csc(x)[/tex]

Step-by-step explanation:

The given expression is

[tex]\frac{\sec(x)}{\tan(x)}[/tex]

We express in terms of basic trigonometric ratios to obtain;

[tex]\frac{\frac{1}{\cos(x)} }{\frac{\sin(x)}{\cos(x)} }[/tex]

This is the same as

[tex]\frac{1}{\cos(x)}\div \frac{\sin(x)}{\cos(x)}[/tex]

[tex]\frac{1}{\cos(x)}\times \frac{\cos(x)}{\sin(x)}[/tex]

Cancel out the common factors;

[tex]\frac{1}{\sin(x)}=\csc(x)[/tex]

Answer:

[tex]\frac{secx}{tanx}[/tex]  = cscx

Step-by-step explanation:

We have given a trigonometric expression.

[tex]\frac{secx}{tanx}[/tex]

We have to simplify the above expression.

Since, we know that

secx is reciprocal of cosx.

secx  =  1/cosx

Tanx is the ratio of sinx and cosx.

Tanx  =  sinx / cosx

Given expression becomes

[tex]\frac{1/cosx}{sinx/cosx}[/tex]

[tex]\frac{1}{cosx}\frac{cosx}{sinx}[/tex]

[tex]\frac{1}{sinx}[/tex]

[tex]\frac{secx}{tanx}[/tex]  = cscx which is the answer.

Final answer:

The division algorithm for polynomials establishes that a polynomial p(x) can be divided by a non-zero polynomial d(x) (with degree less or equal to p(x)), to yield unique quotient q(x) and remainder r(x), with r(x) having a lower degree than d(x).

Explanation:

The division algorithm in the context of polynomials is a fundamental concept in algebra that stipulates for any two polynomials p(x) and d(x), with d(x) ≠ 0 and the degree of d(x) less than or equal to the degree of p(x), there exist unique polynomials q(x) and r(x) such that p(x) = d(x) × q(x) + r(x).

In this scenario, q(x) is referred to as the quotient and r(x) is the remainder. The degree of the remainder r(x) will always be less than the degree of d(x), following the division algorithm.

When applying this algorithm to special sets of polynomials like Legendre polynomials, additional properties can be observed, such as the roots of the polynomials or specific transformations like the Poisson bracket that could arise in mathematical physics. Moreover, the concept extends to rational functions, which are the quotients of polynomials.

Answer:

20

85%

Step-by-step explanation:

You are given the function [tex]S(n)=20\cdot b^n.[/tex]

If n  is the number of hours, then initially n=0 and

[tex]S(0)=20\cdot b^0=20\cdot 1=20.[/tex]

If S(n) is the function of exponential growth, then it can be represented as

[tex]S(n)=I\cdot (1+r)^n,[/tex]

where I is the initial amount, r -is the percent growth rate and n is the number of hours.

If b = 1.85, we can represent it as b = 1 + 0.85. Thus, the hourly percent growth rate of the bacteria would be 0.85=85%.

s(n) = 20b^n

n is the time in hours. At the beginning, the time is zero hours, so n = 0.

s(0) = 20 * b^0

s(0) = 20 * 1

s(0) = 20

The initial amount was 20.

For b = 1.85,

s(n) = 20(1.85)^n

s(n) = 20(1 + 0.85)^n

The hourly growth is 0.85.

0.85 * 100% = 85%

The hourly percent change is 85%.

Answer:

17.85 months

Step-by-step explanation:

original number of rabbits=50

% increase per month=12%

number of months=x

form equation for 1000 rabbits

(112/100) ×50× x = 1000

56x=1000

x=1000/56

x=17.857 months

Answer:

I would say about 60 to 70 pounds. But I have no more information from you to better answer your question, so that's all I have right now.

Step-by-step explanation:

Answer:

x = 25 mm

Step-by-step explanation:

The perimeter of a rectangle is given by

P = 2(l+w) where l is the length and w is the width

We know the perimeter is 60 mm and the width is 5 mm and the length is x

60 = 2(x +5)

Divide each side by 2

60/2 = 2/2(x+5)

30 = x+5

Subtract 5 from each side

30-5 = x+5-5

25 =x

x = 25 mm

[tex]\huge\bold\red{Answer}[/tex]

☑The diagram which is shown above is a rectangle.

✍ Perimeter= 60mm

✍ breadth = 5mm

➡ Perimeter of rectangle = 2(l+b)

✍ 60 = 2(l +5)

✍ 60/2 = l+5

✍ 30 - 5 = l

✍ 25 = l

☑ L = 25mm

❣..hope it helps you..❣

ANSWER

[tex] \angle \: KGI =50 \degree[/tex]

[tex]\angle \: KJI = 130 \degree[/tex]

EXPLANATION

Angles in the same segment are equal.

This implies that;

[tex] \angle \: KGI = \angle \:H[/tex]

Hence,

[tex] \angle \: KGI =50 \degree[/tex]

Also GJKI is a cyclic quadrilateral.

The opposite angles of a cyclic quadrilateral sums up to 180°.

This means that,

[tex] \angle \: KJI + \angle \: KGI = 180 \degree[/tex]

[tex] \angle \: KJI +50 \degree = 180 \degree[/tex]

[tex]\angle \: KJI = 180 \degree - 50 \degree[/tex]

[tex]\angle \: KJI = 130 \degree[/tex]

Our three numbers are...

3 3/10 = 3.3

3.1

3 1/4 = 3.25

So, if we order those from least to greatest, we have...

3.1, 3.25, 3.3

which, in the forms given, is...

3.1, 3-1/4, 3-3/10

Answer:

C. sin(90° - θ)

Step-by-step explanation:

A trig function of an angle equals the cofunction of the angle's complement.

The cofunction of  cosine is sine, and the complement of θ is 90° - θ.

∴ cosθ = sin(90° - θ)

Answer:  Option 'C' is correct.

Step-by-step explanation:

Since we have given that

[tex]\cos \theta[/tex]

We need to find the correct cofunction identity for it.

Cofunction identities represent the relationship among the trigonometric functions.

The value of trigonometric function of an angle is equal to cofunction of its complement.

As we know that sine is a complement of cosine.

so, it becomes,

[tex]\cos \theta=\sin(90^\circ-\theta)[/tex]

Hence, Option 'C' is correct.

Answer:

2.5 %  

Step-by-step explanation:

The simple interest formula is

I = Prt

Data:

I = $75

P = $600

t = 5 yr

Calculation:

75 = 600 × r × 5

75 = 3000 r

Divide each side by 3000

r = 75/3000 = 0.025 = 2.5 % APR

The annual percentage rate is 2.5 %.

Final answer:

The annual rate of simple interest that Remi received from his savings account is 2.5%. This was calculated using the simple interest formula I = PRT, rearranged to solve for R, the annual interest rate.

Explanation:

To work out the annual rate of simple interest that Remi received from his savings account, we can use the formula for simple interest:

I = PRT

where I is the total interest earned, P is the principal amount invested, R is the annual interest rate, and T is the time the money is invested in years.

From the question, we know that Remi invested £600 (P=600), received £75 in interest (I=75), and the investment was for 5 years (T=5). We need to find the annual rate (R).

Rearranging the formula to get value of R, we get:

R = I / (PT)

Substituting the given values:

R = 75 / (600 × 5) (× is the multiplication symbol)

R = 75 / 3000

R = 0.025

To Convert decimal to a percentage, we multiply by 100:

R = 0.025 × 100

R = 2.5%

So, the annual rate of simple interest that Remi received is 2.5%.

Answer:

[tex]3,358\ people[/tex]  

Step-by-step explanation:

The formula to calculate the exponential growth function is equal to

[tex]f(x)=P(e)^{rx}[/tex]  

where  

f(x) is the population  

P is the population in the year 2000  

r is the rate in decimal  

x is number of years since 2000  

e is the mathematical constant number

we have  

[tex]x=2020-2000=20\ years\\ P=2,400\\ r=0.0168[/tex]  

substitute in the formula above  

[tex]f(x)=2,400(e)^{0.0168*20}[/tex]  

[tex]f(x)=3,358\ people[/tex]  

Answer:

see explanation

Step-by-step explanation:

(1)

given

4x² = 81 ( divide both sides by 4 )

x² = [tex]\frac{81}{4}[/tex] ( take the square root of both sides )

x = ± [tex]\sqrt{\frac{81}{4} }[/tex] = ± [tex]\frac{9}{2}[/tex]

---------------------------------------------------------------------

(2)

subtract 7 from both sides

(4x - 3)² = 32 ( take the square root of both sides )

4x - 3 = ± [tex]\sqrt{32}[/tex] = ± 4[tex]\sqrt{2}[/tex]

Add 3 to both sides

4x = 3 ± 4[tex]\sqrt{2}[/tex] ( divide both sides by 4 )

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

Answer:

x     y

-2   7

-1    10.5

0    15.75

1     23.625

2    35.4375

Step-by-step explanation:

The general equation of the exponential function is [tex]y=ab^x[/tex].

We know from our table that when [tex]x=0[/tex], [tex]y=15.75[/tex]. Let's replace those values in our equation:

[tex]y=ab^x[/tex]

[tex]15.75=ab^0[/tex]

Remember that [tex]b^0=1[/tex], so:

[tex]15.75=a(1)[/tex]

[tex]15.75=a[/tex]

[tex]a=15.75[/tex]

We also know from our table that when [tex]x=-1[/tex], [tex]y=10.5[/tex]. Let's replace the values again:

[tex]y=ab^x[/tex]

[tex]10.5=ab^{-1}[/tex]

But we now know that [tex]a=15.75[/tex], so let's replace that value as well:

[tex]10.5=15.75b^{-1}[/tex]

Remember that [tex]b^{-1}=\frac{1}{b}[/tex], so:

[tex]10.5=\frac{15.75}{b}[/tex]

[tex]10.5b=15.75[/tex]

[tex]b=\frac{15.75}{10.5}[/tex]

[tex]b=1.5[/tex]

Now, we can put it all together to complete our exponential function:

[tex]y=ab^x[/tex]

[tex]y=15.75(1.5)^x[/tex]

To find the missing values, we just need to evaluate our function at [tex]x=1[/tex] and [tex]x=2[/tex]:

- For [tex]x=1[/tex]

[tex]y=15.75(1.5)^x[/tex]

[tex]y=15.75(1.5)^1[/tex]

[tex]y=23.625[/tex]

- For [tex]x=2[/tex]

[tex]y=15.75(1.5)^x[/tex]

[tex]y=15.75(1.5)^2[/tex]

[tex]y=35.4375[/tex]

Answer:

Step-by-step explanation:

The ratios of corresponding linear dimensions are identical:

  5/2 = 15/6 = 2.5

so we can conclude the figures are similar. The scale factor is the ratio of corresponding dimensions:

  first : second = 5 : 2 . . . or . . . 2.5 : 1

Answer:

Yes.

Scale factor = 2.5.

Step-by-step explanation:

They are similar because corresponding dimensions are in the same ratio:

2/5 = 6/15.

The scale factor = 5/2 = 2.5.

Answer: -0.6

Step-by-step explanation:

First thing to do is to solve for θ and ϕ from the given information

tan(θ) = 4/3,

θ = tan-¹4/3,

θ = 53.1°

Since tan is positive in quadrant III, θ = 53.1°

Also,

sin(ϕ) = − 10/10 ,

ϕ = sin-¹-1

ϕ = 270°

If ϕ is in the fourth quadrant, that gives 360 - ϕ i.e 360 - 270 = 90°

Substituting the values of θ and ϕ into sin(θ − ϕ), we have;

Sin(53.1 - 90)

= sin (-36.9°)

= -0.6

The given expression sin(θ - ϕ), with tan(θ) = 4/3, θ in Quadrant III, sin(ϕ) = -10/10, and ϕ in Quadrant IV, is evaluated by using trigonometric principles and identities. Upon calculations, sin(θ - ϕ) comes out to be -3/5.

The question is asking us to evaluate the expression sin(θ − ϕ), given that tan(θ) = 4/3, θ is in Quadrant III, sin(ϕ) = -10/10, and ϕ is in Quadrant IV. In trigonometry, tan θ = sin θ/cos θ. We have tan θ = 4/3 and we know that in Quadrant III, tangent is positive but sine and cosine are negative. So, we can make a right triangle where the opposite side is 4 (basing this on the absolute value of the tan θ) and the adjacent side is 3. The hypotenuse then, by using Pythagoras theorem, comes out to be 5. Then sin θ = -4/5 and cos θ = -3/5.

For sin(ϕ), we are given that it equals -1. In Quadrant IV, sine is negative and cosine is positive, so cos ϕ = √(1 - (-1)^2) = 0.

Finally, utilizing the formula sin (a ± ß) = sin a cos ß ± cos a sin ß, we plug in our values to come to the solution sin(θ - ϕ) = (sin θ cos ϕ) - (cos θ sin ϕ) = ((-4/5)*0) - ((-3/5)*-1) = -3/5

https://brainly.com/question/11016599

#SPJ11

a) Yes, s is a subspace of v because the three conditions for a subspace are satisfied. b) Yes, s is a subspace of v because the three conditions for a subspace are satisfied. c) Yes, s is a subspace of v because the three conditions for a subspace are satisfied.

a. v=mn(r) is the vector space of all m x n matrices with entries in the real numbers. In this case, s is a subspace of v if it satisfies the three conditions: (1) the zero vector is in s, (2) s is closed under vector addition, and (3) s is closed under scalar multiplication. Since the zero matrix is an upper triangular matrix, it satisfies the first condition. If two upper triangular matrices are added, the result will also be an upper triangular matrix, satisfying the second condition. And if an upper triangular matrix is multiplied by a scalar, the result is still an upper triangular matrix, satisfying the third condition. Therefore, s is a subspace of v.

b. In this case, v is the vector space of all real-valued functions defined on the interval (-∞, ∞). The subset s consists of functions satisfying f(0) = 0. Similar to part a, we need to check if s satisfies the three conditions to be considered a subspace. The zero function satisfies f(0) = 0, so it satisfies the first condition. If two functions in s are added, the sum will also have f(0) = 0, satisfying the second condition. And if a function in s is multiplied by a scalar, the result will still have f(0) = 0, satisfying the third condition. Therefore, s is a subspace of v.

c. In this case, v is the vector space of all second-order linear homogeneous differential equations. The subset s consists of functions satisfying y'' - 4y' + 3y = 0. To be considered a subspace, s would need to satisfy the three conditions: (1) the zero element is in s, (2) s is closed under addition, and (3) s is closed under scalar multiplication. The zero function satisfies the differential equation, so it satisfies the first condition. If two functions in s are added, the sum will also satisfy the differential equation, satisfying the second condition. And if a function in s is multiplied by a scalar, the result will still satisfy the differential equation, satisfying the third condition. Therefore, s is a subspace of v.

Answer:

Final answer is [tex]x=\frac{\pi}{3}[/tex] and [tex]x=\frac{5\pi}{3}[/tex].

Step-by-step explanation:

Given equation is [tex]1-\cos\left(\theta\right)=\frac{1}{2}[/tex]

Now we need to find the solution of  [tex]1-\cos\left(\theta\right)=\frac{1}{2}[/tex] in given interval [tex][0, 2\pi ][/tex].

[tex]1-\cos\left(\theta\right)=\frac{1}{2}[/tex]

[tex]-\cos\left(\theta\right)=\frac{1}{2}-1[/tex]

[tex]-\cos\left(\theta\right)=-\frac{1}{2}[/tex]

[tex]\cos\left(\theta\right)=\frac{1}{2}[/tex]

which gives [tex]x=\frac{\pi}{3}[/tex] and [tex]x=\frac{5\pi}{3}[/tex] in the given interval.

Hence final answer is [tex]x=\frac{\pi}{3}[/tex] and [tex]x=\frac{5\pi}{3}[/tex].

In the interval [0, 2π], the cosine function takes on the value 1/2 at two specific angles: π/3 and 5π/3.

To solve the equation 1 - cos(theta) = 1/2 on the interval [0, 2π], this is done by:

Subtract 1/2 from both sides of the equation to isolate the cosine term:

1 - cos(theta) - 1/2 = 0

-cos(theta) + 1/2 = 0

So multiply both sides by -1 to get rid of the negative sign: cos(theta) - 1/2 = 0

1/2 can be written as 2/4: cos(theta) - 2/4 = 0

So look for common denominator for the fraction on the left side, and is 4:

(4*cos(theta) - 2)/4 = 0

Then Multiply both sides by 4 to remove the fraction: 4*cos(theta) - 2 = 0

So, add 2 to both sides: 4*cos(theta) = 2

Lastly divide both sides by 4: cos(theta) = 1/2

Therefore, the solutions to the equation are: theta = π/3 and theta = 5π/3.

Read more about  interval  here:

https://brainly.com/question/30913519

#SPJ6

Answer:

the answer is D. 3^7√3

Step-by-step explanation:

For this case we have to define properties of powers and roots:

[tex]\sqrt [n] {a ^ m} = a ^ {\frac {m} {n}}[/tex]

We also have to:

[tex]3 ^ {15} = 3 ^ {14} * 3[/tex]

So:

[tex]\sqrt {3 ^ {15}} = \sqrt {3 ^ {14} * 3} = 3 ^ {\frac {14} {2}} * \sqrt {3} = 3 ^ 7 * \sqrt {3}[/tex]

Answer:

[tex]3 ^ 7 * \sqrt {3}[/tex]

Option D