An arc has a length if 20π and a measure of 30°., №18009929, 02.04.2021 03:04

Mathematics

An arc has a length if 20π and a measure of 30°. What js the radius of the circle?

Step-by-step answer

P Answered by Master

r=120

Step-by-step explanation:

An arc has a length if 20π and a measure of 30°. What js the radius of the circle?

Mathematics

Consider circle h with a 3 centimeter radius. if the length of minor arc rt is 2 3 π, what is the measure of ∠rst? a) 20° b) 30° c) 40° d) 60°

Step-by-step answer

P Answered by Master

If the arc is ⅔π, then your angle will be none of these. The central angle (if s is the center I presume) would be 120, since π is 180 and ⅔ of 180 is 120. The only time you would get one of those choices is if the angle went not to the center but straight across to the other side, in that case it would be 60.
Consider circle h with a 3 centimeter radius. if the length of minor arc rt is 2 3 π, what is the me

Consider circle h with a 3 centimeter radius. if the length of minor arc rt is 2 3 π, what is the me

Mathematics

Consider circle h with a 3 centimeter radius. if the length of minor arc rs is 5/2π, what is the measure of ∠rst? assume that chords rs and ts are congruent. a. 20 degrees b. 25 degrees c. 30 degrees d. 40 degrees

Step-by-step answer

P Answered by PhD

4

the picture in the attached figure N 1
the answer in the attached figure N 2

Consider circle h with a 3 centimeter radius. if the length of minor arc rs is 5/2π, what is the mea

Mathematics

Consider the circle with a 4 centimeter radius. if the length of major arc rt is 64/9π, what is the measure of ∠rst? a.20 b.25 c.30 d.40

Step-by-step answer

P Answered by PhD

3

Option D is correct.i.e., ∠ RST = 40°

Step-by-step explanation:

Given: Radius of circle, r = 4 cm

Center of circle is S and Length of Major ARC RT = $\frac{64}{9}\times\pi$

To find: ∠ RST

Figure is attached

Let $\theta$ be the ∠ RST.

we use the formula of Length of Arc,

$Length\: of\: Arc\: =\:\frac{\theta}{360^{\circ}}\times2\pi r$

We use this formula to find the angle made by arc RT at centre S,

$Length\:of\:major\:Arc\:=\:\frac{360^{\circ}-\theta}{360^{\circ}}\times2\pi r$

$\frac{64}{9}\times\pi=\:\frac{360^{\circ}-\theta}{360^{\circ}}\times2\pi \times4$

$\frac{64}{9}\times\pi=\:\frac{360^{\circ}-\theta}{360^{\circ}}\times8\pi$

$\frac{360^{\circ}-\theta}{360^{\circ}}=\frac{64\times\pi}{9\times8\pi}$

$\frac{360^{\circ}-\theta}{360^{\circ}}=\frac{8}{9}$

$360^{\circ}-\theta=\frac{8\times360}{9}$

$360^{\circ}-\theta=8\times40$

$360^{\circ}-\theta=320$

$-\theta=320-360$

$-\theta=-40$

$\theta=40$

Therefore, Option D is correct.i.e., ∠ RST = 40°

Consider the circle with a 4 centimeter radius. if the length of major arc rt is 64/9π, what is the

Mathematics

2. a ski resort is building a new ski lift that will transport tourists from the base of the mountain to its highest point. this mountain has a vertical height of 200 yards, and the ski lift will rise at an angle of 40 degrees. when the project is completed, how many yards, d, will a tourist travel from the base of the mountain to its peak? part i: sketch a figure to illustrate the scenario above. label the vertices and the lengths that are given in the question. (3 points) part ii: using your sketch from part i, write an equation using a trigonometric

Step-by-step answer

P Answered by Master

36

You are given the mountain that has a vertical height of 200 yards, and the ski lift will rise at an angle of 40 degrees. You are asked to find how many yards will a tourist travel from the base of the mountain to its peak. You can solve this using trigonometric function. More importantly, imagine that the mountain has a 90 - degree vertical height so that when you connect the peak of the mountain down to its base and then connected to the ski, you can form a right triangle.

Use the sine function.
sine β = opposite side / hypotenuse side
the opposite side will be the height of the mountain and the hypotenuse side will be the distance of the ski from the bottom going to the peak of the mountain

sine 40° = 100 yards / hypotenuse side
hypotenuse side = 100 yards / sine 40°
hypotenuse side = 156 yards

Mathematics

16. photon lighting company determines that the supply and demand functions for its most popular lamp are as follows: s(p) = 400 - 4p + 0.2p4 and d(p) = 2,800 - 0.0012p3, where p is the price. determine the price for which the supply equals the demand. a) $93.24 b) $100.24 c) $96.24 d) $99.24 17. write a polynomial function of minimum degree with real coefficients whose zeros include those listed. write the polynomial in standard form. 2, -4, and 1 + 3i (1 point) a) f(x) = x4 - 2x2 + 36x - 80 b) f(x) = x4 - 3x3 + 6x2 - 18x + 80 c) f(x) = x4 - 9x2

Step-by-step answer

P Answered by PhD

80

QUESTION 16

If the demand and supply are equal, then we equate the two functions in p and solve for p.

That is

S(p)=D(p)

$400 - 4p + + 0.00002 {p}^{4} = 2800 - 0.0012 {p}^{3}$

We can rearrange to obtain,

$0.00002 {x}^{4} + 0.0012 {x}^{3} - 4x - 2400 = 0$

$2 {p}^{2} + 1200 {p}^{3} - 400000p - 240000000 = 0$

The real roots of this polynomial equation are:

$p = - 118.26 \: p = 96.24$

Since price can not be negative, we discard the negative value ,

p = 96.24

The correct answer for question 16 is C.

QUESTION 17

We were given the solution to this polynomial as

x=2,x=-4, x=1+3i

We need to recognize the presence of the complex root and treat it nicely.

There is one property about complex roots of polynomial equations called the complex conjugate property. According to this property, if

a + bi

is a solution to

p(x)

then the complex conjugate

a - bi

is also a root.

Since

x = 1 + 3i

is a solution then,

x = 1 - 3i

is also a solution.

Therefore we have

f(x) = (x - 2)(x + 4)(x - (1+3i))(x - (1 - 3i)

$f(x) = ( {x}^{2} + 4x - 2x - 8)( {x}^{2} - (1 - 3i)x - (1 + 3i)x + (1 + 3i)(1 - 3i))$

We expand to obtain,

$( {x}^{2} + 4x - 2x - 8)( {x}^{2} - x + 3xi - x - 3xi + 1 + 9)$

Note that:

${i}^{2} = - 1$

$f(x) = ( {x}^{2} + 2x - 8)( {x}^{2} - 2x + 10)$

We now expand to obtain,

$f(x) = {x}^{4} - 2 {x}^{3} + 10 {x}^{2} + 2 {x}^{3} - 4 {x }^{2} + 20x - 8 {x}^{2} + 16x - 80$

We simplify further to obtain,

$f(x) = {x}^{4} + 2 {x}^{2} + 36x - 80$

The correct answer for question 17 is A.

QUESTION 18

If

-2i

is a zero of the polynomial,

f(x)=x^4-45x^2-196

then the complex conjugate

is also a zero,

This means that ,

(x+2i), (x-2i)

are factors of the polynomial.

The product of these two factors,

(x+2i), (x-2i) =x^2-(2i)^2=x^2+4

is also a factor , so we use it to divide and get the remaining factors.

see diagram for long division.

The above polynomial can therefore factored completely as,

f(x)=(x^2+4)(x^2-49)

Applying our knowledge from difference of two squares, we obtain,

f(x)=(x+2i)(x-2i))(x-7)(x+7)

Hence all the zeroes of these polynomial can be found by setting

(x+2i)(x-2i))(x-7)(x+7)=0

This implies,

x=-2i,x=2i,x=7,x=-7

The correct answer for question 18 is B

QUESTION 19

We were asked to find the horizontal and vertical asymptote of

$f(x)=\frac{2x^2+1}{x^2-1}$

To find the horizontal asymptote, divide the term with the highest degree in the numerator by the term with the highest degree in the denominators. That is the horizontal asymptote is given by,

$y=\frac{2x^2}{x^2}=2$

For vertical asymptote, equate the denominator to zero and solve for x.

x^2-1=

$\Rightarrow x=-1, x=1$

None of the options is correct, so the correct answer for question 19 is A.

QUESTION 20

We are converting,

$\frac{5\pi}{6}$

to degrees .

To convert from radians to degrees, multiply by,

$\frac{180\degree}{\pi}$

That is,

$\frac{5\pi}{6}=\frac{5\pi}{6} \times \frac{180\degree}{\pi}$

We simplify to obtain,

$\frac{5\pi}{6}=\frac{5}{1} \times \frac{180\degree}{1}=5\times30\degree =150\degree$

The correct answer is B.

QUESTION 21

Recall the mnemonics, SOH CAH TOA

The sine ratio is given by,

$\sin(B)=\frac{21}{75}=\frac{7}{25}$

From the diagram,

$\tan(B)=\frac{21}{72}=\frac{7}{24}$

The correct answer is C.

QUESTION 22

From the above diagram, We can determine the value of x using the sine or cosine ratio, depending on where the 17 is placed.

Using the cosine ratio, we obtain,

$\cos(58\degree)=\frac{17}{x}$

We can simply switch positions to make x the subject.

$x=\frac{17}{\cos(58\degree)}$

$x=\frac{17}{0.5299}$

x=32.08

Hence the correct answer is A.

QUESTION 23

Coterminal angles have the same terminal sides.

To find coterminal angles, we keep adding or subtracting 360 degrees.

See diagram.

$x=202\degree$

is coterminal with

$202\degree +360\degree =562\degree$

or

$202\degree -360\degree =-158\degree$

The correct answer is D.

See the attached file for continuation.

16. photon lighting company determines that the supply and demand functions for its most popular lam

Mathematics

16. photon lighting company determines that the supply and demand functions for its most popular lamp are as follows: s(p) = 400 - 4p + 0.2p4 and d(p) = 2,800 - 0.0012p3, where p is the price. determine the price for which the supply equals the demand. a) $93.24 b) $100.24 c) $96.24 d) $99.24 17. write a polynomial function of minimum degree with real coefficients whose zeros include those listed. write the polynomial in standard form. 2, -4, and 1 + 3i (1 point) a) f(x) = x4 - 2x2 + 36x - 80 b) f(x) = x4 - 3x3 + 6x2 - 18x + 80 c) f(x) = x4 - 9x2

Step-by-step answer

P Answered by PhD

80

QUESTION 16

If the demand and supply are equal, then we equate the two functions in p and solve for p.

That is

S(p)=D(p)

$400 - 4p + + 0.00002 {p}^{4} = 2800 - 0.0012 {p}^{3}$

We can rearrange to obtain,

$0.00002 {x}^{4} + 0.0012 {x}^{3} - 4x - 2400 = 0$

$2 {p}^{2} + 1200 {p}^{3} - 400000p - 240000000 = 0$

The real roots of this polynomial equation are:

$p = - 118.26 \: p = 96.24$

Since price can not be negative, we discard the negative value ,

p = 96.24

The correct answer for question 16 is C.

QUESTION 17

We were given the solution to this polynomial as

x=2,x=-4, x=1+3i

We need to recognize the presence of the complex root and treat it nicely.

There is one property about complex roots of polynomial equations called the complex conjugate property. According to this property, if

a + bi

is a solution to

p(x)

then the complex conjugate

a - bi

is also a root.

Since

x = 1 + 3i

is a solution then,

x = 1 - 3i

is also a solution.

Therefore we have

f(x) = (x - 2)(x + 4)(x - (1+3i))(x - (1 - 3i)

$f(x) = ( {x}^{2} + 4x - 2x - 8)( {x}^{2} - (1 - 3i)x - (1 + 3i)x + (1 + 3i)(1 - 3i))$

We expand to obtain,

$( {x}^{2} + 4x - 2x - 8)( {x}^{2} - x + 3xi - x - 3xi + 1 + 9)$

Note that:

${i}^{2} = - 1$

$f(x) = ( {x}^{2} + 2x - 8)( {x}^{2} - 2x + 10)$

We now expand to obtain,

$f(x) = {x}^{4} - 2 {x}^{3} + 10 {x}^{2} + 2 {x}^{3} - 4 {x }^{2} + 20x - 8 {x}^{2} + 16x - 80$

We simplify further to obtain,

$f(x) = {x}^{4} + 2 {x}^{2} + 36x - 80$

The correct answer for question 17 is A.

QUESTION 18

If

-2i

is a zero of the polynomial,

f(x)=x^4-45x^2-196

then the complex conjugate

is also a zero,

This means that ,

(x+2i), (x-2i)

are factors of the polynomial.

The product of these two factors,

(x+2i), (x-2i) =x^2-(2i)^2=x^2+4

is also a factor , so we use it to divide and get the remaining factors.

see diagram for long division.

The above polynomial can therefore factored completely as,

f(x)=(x^2+4)(x^2-49)

Applying our knowledge from difference of two squares, we obtain,

f(x)=(x+2i)(x-2i))(x-7)(x+7)

Hence all the zeroes of these polynomial can be found by setting

(x+2i)(x-2i))(x-7)(x+7)=0

This implies,

x=-2i,x=2i,x=7,x=-7

The correct answer for question 18 is B

QUESTION 19

We were asked to find the horizontal and vertical asymptote of

$f(x)=\frac{2x^2+1}{x^2-1}$

To find the horizontal asymptote, divide the term with the highest degree in the numerator by the term with the highest degree in the denominators. That is the horizontal asymptote is given by,

$y=\frac{2x^2}{x^2}=2$

For vertical asymptote, equate the denominator to zero and solve for x.

x^2-1=

$\Rightarrow x=-1, x=1$

None of the options is correct, so the correct answer for question 19 is A.

QUESTION 20

We are converting,

$\frac{5\pi}{6}$

to degrees .

To convert from radians to degrees, multiply by,

$\frac{180\degree}{\pi}$

That is,

$\frac{5\pi}{6}=\frac{5\pi}{6} \times \frac{180\degree}{\pi}$

We simplify to obtain,

$\frac{5\pi}{6}=\frac{5}{1} \times \frac{180\degree}{1}=5\times30\degree =150\degree$

The correct answer is B.

QUESTION 21

Recall the mnemonics, SOH CAH TOA

The sine ratio is given by,

$\sin(B)=\frac{21}{75}=\frac{7}{25}$

From the diagram,

$\tan(B)=\frac{21}{72}=\frac{7}{24}$

The correct answer is C.

QUESTION 22

From the above diagram, We can determine the value of x using the sine or cosine ratio, depending on where the 17 is placed.

Using the cosine ratio, we obtain,

$\cos(58\degree)=\frac{17}{x}$

We can simply switch positions to make x the subject.

$x=\frac{17}{\cos(58\degree)}$

$x=\frac{17}{0.5299}$

x=32.08

Hence the correct answer is A.

QUESTION 23

Coterminal angles have the same terminal sides.

To find coterminal angles, we keep adding or subtracting 360 degrees.

See diagram.

$x=202\degree$

is coterminal with

$202\degree +360\degree =562\degree$

or

$202\degree -360\degree =-158\degree$

The correct answer is D.

See the attached file for continuation.

Mathematics

2. a ski resort is building a new ski lift that will transport tourists from the base of the mountain to its highest point. this mountain has a vertical height of 200 yards, and the ski lift will rise at an angle of 40 degrees. when the project is completed, how many yards, d, will a tourist travel from the base of the mountain to its peak? part i: sketch a figure to illustrate the scenario above. label the vertices and the lengths that are given in the question. (3 points) part ii: using your sketch from part i, write an equation using a trigonometric

Step-by-step answer

P Answered by Master

36

You are given the mountain that has a vertical height of 200 yards, and the ski lift will rise at an angle of 40 degrees. You are asked to find how many yards will a tourist travel from the base of the mountain to its peak. You can solve this using trigonometric function. More importantly, imagine that the mountain has a 90 - degree vertical height so that when you connect the peak of the mountain down to its base and then connected to the ski, you can form a right triangle.

Use the sine function.
sine β = opposite side / hypotenuse side
the opposite side will be the height of the mountain and the hypotenuse side will be the distance of the ski from the bottom going to the peak of the mountain

sine 40° = 100 yards / hypotenuse side
hypotenuse side = 100 yards / sine 40°
hypotenuse side = 156 yards

Mathematics

Which statements about the clock are accurate? Select three options. The central angle formed when one hand points at 1 and the other hand points at 3 is 30°. The circumference of the clock is approximately 62.8 inches. The minor arc measure when one hand points at 12 and the other hand points at 4 is 120°. The length of the major arc between 3 and 10 is approximately 31.4 inches. The length of the minor arc between 6 and 7 is approximately 5.2 inches.

Step-by-step answer

P Answered by PhD

22

Option 2, 3 and 5 are correct.

Step-by-step explanation:

Given question is incomplete, here is the complete question.

The face of a clock is divided into 12 equal parts.

Which statements about the clock are accurate? Select three options.

The central angle formed when one hand points at 1 and the other hand points at 3 is 30°.

The circumference of the clock is approximately 62.8 inches.

The minor arc measure when one hand points at 12 and the other hand points at 4 is 120°.

The length of the major arc between 3 and 10 is approximately 31.4 inches.

The length of the minor arc between 6 and 7 is approximately 5.2 inches.

Option 1. "Central angle when one hand points at 1 and the other hand points at 3."

Since, face of the clock has been divided in 12 equal parts.

So, central angle at the center for each part = $\frac{360}{12}$

= 30°

Number of parts between 1 and 3 = 3

So central angle between 1 and 3 = 3 × 30°

= 90° (False)

Option 2. "Circumference of the clock = 62.8° inches."

Formula to be used

Circumference = 2πr of a circle

r = radius of the circle

Therefore circumference of he circle = 2π(10)

= 20π

= 62.8 inches (True)

Option 3. "Minor arc measure when one hand points at 12 and the other hand points at 4 is 120°."

Number of parts between 12 and 4 = 4

So Central angle formed = 4 × 30°

= 120° (True)

Option 4. "Length of major are between 3 and 10 = 31.4 inches."

Number of parts between 3 and 10 = 7

Central angle formed by hands = 7 × 30°

= 210°

$\text{Length of arc}=\frac{\pi }{360{^\circ}} \times (2\pi r)$

$=\frac{210}{360}\times (2\pi )(10)$

= 39.65 inches (False)

Option 5. "The length of the minor arc between 6 and 7 is approximately 5.2 inches."

Number of parts between 6 and 7 = 1

Central angle between 6 and 7 = 30°

Length of minor arc = $\frac{30}{360}\times (2\pi )(10)$

$=\frac{20\pi }{12}=5.24$

≈ 5.2 inches (True)