Thomas has constant speeds for both running and walking. When a down-escalator is moving, Thomas can run down it in
15
seconds or walk down it in
30
seconds. One day, when the escalator was broken (and stationary), it took Thomas
20
seconds to run down it.

How long, in seconds, would it take Thomas to walk down the broken escalator?

Answer:

Step-by-step explanation:

Assumed Knowledge

Motivation

Content

Radii and chords

Angles in a semicircle

Angles at the centre and circumference

Cyclic quadrilaterals

The alternate segment theorem

Similarity and Circles

Links Forward

Converse of the circle theorems

Coordinate geometry

Calculus

History and Applications

The Euler line

The nine-point circle

Morley’s trisector theorem

Appendix − Converses to the Circle Theorems

Answers to Exercises

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ASSUMED KNOWLEDGE

Introductory plane geometry involving points and lines, parallel lines and transversals, angle sums of triangles and quadrilaterals, and general angle-chasing.

Experience with a logical argument in geometry written as a sequence of steps, each justified by a reason.

Ruler-and-compasses constructions.

The four standard congruence tests and their application to proving properties of and tests for special triangles and quadrilaterals.

The four standard similarity tests and their application.

Trigonometry with triangles.

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MOTIVATION

Most geometry so far has involved triangles and quadrilaterals, which are formed by intervals on lines, and we turn now to the geometry of circles. Lines and circles are the most elementary figures of geometry − a line is the locus of a point moving in a constant direction, and a circle is the locus of a point moving at a constant distance from some fixed point − and all our constructions are done by drawing lines with a straight edge and circles with compasses. Tangents are introduced in this module, and later tangents become the basis of differentiation in calculus.

The theorems of circle geometry are not intuitively obvious to the student, in fact most people are quite surprised by the results when they first see them. They clearly need to be proven carefully, and the cleverness of the methods of proof developed in earlier modules is clearly displayed in this module. The logic becomes more involved − division into cases is often required, and results from different parts of previous geometry modules are often brought together within the one proof. Students traditionally learn a greater respect and appreciation of the methods of mathematics from their study of this imaginative geometric material.

The theoretical importance of circles is reflected in the amazing number and variety of situations in science where circles are used to model physical phenomena. Circles are the first approximation to the orbits of planets and of their moons, to the movement of electrons in an atom, to the motion of a vehicle around a curve in the road, and to the shapes of cyclones and galaxies. Spheres and cylinders are the first approximation of the shape of planets and stars, of the trunks of trees, of an exploding fireball, and of a drop of water, and of manufactured objects such as wires, pipes, ball-bearings, balloons, pies and wheels.

a) The series of transformations to map Circle A onto Circle B is likely a translation. b) The series of transformations to map Circle C onto Circle A is a translation, rotation, and scaling.

a. To map Circle A onto Circle B, we must determine the transformation sequence. We can scale, rotate, and translate.

Translation: An object is translated horizontally or vertically without affecting shape or orientation. To match Circle B, Circle A appears to need to be shifted right and up.

Rotation: An object rotates around a fixed point. Rotation may not be needed since Circle A and Circle B appear to be aligned.

Scaling: An object is resized by scaling. Since Circle A and Circle B are the same size, scaling doesn't seem necessary.

Thus, Circle A-to-Circle B transformations are likely translations.

b. Again, we must determine the transformation sequence to transfer Circle C onto Circle A. Translation: Based on their relative positions, Circle C needs to be shifted left and down to match Circle A.

Rotation: To align Circles A and C, they must be rotated.

Scaling: Circle A is larger than Circle C, hence Circle C should be scaled down.

Thus, translating, rotating, and scaling map Circle C onto Circle A.

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Answer:

320

Step-by-step explanation:

400*80/100=320

Answer:

$320 is the final price, $80 is the amount taken off

Answer:

n= -9, t= -9

Step-by-step explanation:

1) -1/3(n+15)=-2

-1/3×n+-1/3×15=-2

-1/3×n -5= -2

-1/3 n-5+5= -2+5

-1/3×n=3

3/-1×-1/3×n=3/-1×3

n= -9

2) 46 = -6t -8

−6t−8=46

−6t−8+8=46+8

−6t=54

Divide both sides by -6

t=-9

Answer: angle 5 and angle 10

angle 8 and angle 11

hope this helps!

Answer:

[tex]\frac{1}{3x^{\frac{4}{5}}}[/tex]

Step-by-step explanation:

Subtract the exponents: 3/5 - 7/5 = -4/5

Equation would become 3^-4/5

Which is equivalent to 1/3x^4/5

Step-by-step explanation:

x²+3x+4=0

x²+3x_ +2=0

x²+3x_2=-2

x²+3x_2+(3x_4)²=-2+(3/4)²

(x+3/4) =-2+9_16

x+3_4 = -32+9__16 =√-23_6

x+3_4 =-√23_4

x = -3+√-23___4

x = -3- √-23___4 , -3+√-23___4 //

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Answer:

can i see the picture?

Answer:

I think it is K = 2

Step-by-step explanation:

Hope this helps.

Answer:

m=E/c².

Step-by-step explanation:

Transposing the formula

E=mc²

we divide both sides by c² then cancel out where necessary so m would stand alone

E/c²=mc²/c²

cancelling c² on the right hand side, equation becomes

m= E/c²

Answer:

50%

Step-by-step explanation:

45 is half of 90. A percent is out of 100. So when you split 100 in half, it equals 50. The answer is 50%. Glad I could help!

Answer:

50% of the students voted for the athlete.

Step-by-step explanation:

half of 90 = 45 which means 45 + 45 = 90

so 50% of the class voted for the athlete

Answer: or 13%

Step-by-step explanation:

Answer: D.

Step-by-step explanation:

Answer:

50.265482457436 inches

Step-by-step explanation:

radius = 8 inches

C = 2×pi×r

= 2×pi×8

C = 50.265482457436 inches

Answer:

The first one

Step-by-step explanation:

The amplitude is usually the first number (for example, in this equation, the 3). Then you also know that the right or left shift depends on the addition or subtraction sign. Usually, the subtraction sign means to the right and the addition sign means to the left.

We have that the graph of which function has an amplitude of 3 and a right phase shift of [tex]\phi= \frac{\pi}{4}[/tex]  is

[tex]3sin2(x+\phi)[/tex]

The Correct option is D

From the Question we are told that

A graph has

Amplitude of 3

Phase [tex]\phi= \frac{\pi}{4}[/tex]

Generally the equation for the Wave Equation  is mathematically given as

[tex]Asin(kx-wt+\phi)[/tex]

Therefore the Equation for this particular wave is

[tex]3sin2(x+\phi)[/tex]

Hence

The graph of which function has an amplitude of 3 and a right phase shift of [tex]\phi= \frac{\pi}{4}[/tex]  is

[tex]3sin2(x+\phi)[/tex]

The Correct option is D

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9514 1404 393

Answer:

Step-by-step explanation:

The angle marks show that ΔRST is isosceles, so RT = RS. They also show ΔRTU to be equilateral, so UT = RT.

  UT = RS

  x -5 = 15

  x = 20 . . . . . . add 5

__

ΔRTU is equilateral, so ∠RTU is 60°. The angle marked 3y° is complementary to that.

  3y = 30

  y = 10 . . . . . divide by 3

[tex]\color{plum}\tt \: D) 38.5 [/tex]

Diameter of the stained glass window Joey needs = 3.5

Area of the window = quantity of glass Joey needs

Area of the window :

[tex] = \tt\pi {r}^{2} [/tex]

[tex] = \tt3.14 \times 3.5 \times 3.5[/tex]

[tex] =\tt 3.14 \times 12.25[/tex]

[tex] =\tt 38.46[/tex]

38.46 can be rounded off to 38.5.

Therefore, quantity of glass needed to make the stained glass window = 38.5

Answer:

the area is 670.32 but it's not in there

Answer:

y=x+5

Step-by-step explanation:

y increases by 5 so in y=mx+b b=5 and x increases by 1 so still x thus the answer is y=x+5

Answer:

Step-by-step explanation:

therefore

[tex]r = \frac{3 \frac{1}{4} }{2} [/tex]

[tex]r = \frac{ \frac{13}{2} }{2} [/tex]

[tex]r = \frac{13}{2} \div 2[/tex]

[tex]r = \frac{13}{4} [/tex]

[tex]area \: of \: a \: circle \: = \pi {r}^{2} [/tex]

[tex] = \pi (\frac{13}{4} ) ^{2} i {n}^{2} [/tex]

[tex] = \pi \times \frac{169}{16} {in}^{2} [/tex]

[tex] = \frac{169\pi}{16} i {n}^{2} [/tex]

[tex] = 33.2 \: i {n}^{2} [/tex]

Answer: The student needs 144 points to get a score of 96%

Step-by-step explanation:

Answer:

C

Step-by-step explanation:

First find the radius:

5 pi = 2 pi r

cancel out pi

5 = 2r

r = 5/2

Then find the area:

A = pi r^2

A = pi 5/2^2

A = 25/4 pi

The amount of ribbon needed for a square poster will be 20.

And, The amount of ribbon needed for a triangular poster will be 15.

What is an expression?

Mathematical expression is defined as the collection of the numbers variables and functions by using operations like addition, subtraction, multiplication, and division.

Given that;

The system of equation are;

⇒ s = t + 5

⇒ 45 + 2t = 110​

Now,

Since, The system of equation are;

⇒ s = t + 5         ..(i)

⇒ 4s + 2t = 110​  ..(ii)

Solve equation (ii) for t as;

⇒ 45 + 2t = 110​

⇒ 4s + 2t - 4s = 110​ - 4s

⇒ 2t = 110 - 4s

⇒ t = 55 - 2s

Substitute above value in equation (i), we get;

⇒ s = t + 5  

⇒ s = 55 - 2s + 5  

⇒ s + 2s = 60

⇒ 3s = 60

⇒ s = 20

And,   ⇒ t = 55 - 2s

⇒ t = 55 - 2 x 20

⇒ t = 55 - 40

⇒ t = 15

Thus, The amount of ribbon needed for a square poster will be 20.

And, The amount of ribbon needed for a triangular poster will be 15.

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Answer:

c=-16

Step-by-step explanation:

Answer:

(b) is correct 9,13,17,21,25

Answer:

14.6

Step-by-step explanation:

difference: 12-7=5

division: 73/5=14.6

The 73 divided by the difference between 12 and 7 is equal to 14.6.

To properly calculate 73 divided by the difference between 12 and 7, we follow these steps:

Calculate the difference between 12 and 7:

12 - 7 = 5

Divide 73 by the difference obtained:

73 ÷ 5 = 14.6

Therefore, 73 divided by the difference between 12 and 7 is equal to 14.6. This means that if we take the value 73 and divide it by the difference between 12 and 7, the quotient will be 14.6. The difference between 12 and 7 is 5, and when we divide 73 by 5, we obtain 14.6. It is important to note that the result is a decimal value, specifically 14.6.

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Answer:

Yes

Step-by-step explanation:

[tex]4x + 7 + 6x - 1\\10x + 6[/tex]

[tex]15x + 5 + 1 -5x\\10x+6[/tex]

They both equal 10x + 6

Answer:

no they are not equal bcz Lhs is not =Rhs

Answer:

The numbers are  x= 9 and y =6

The product of two numbers

x y  = 9 × 6 = 54

Step-by-step explanation:

Step(i):-

Let x and y are two numbers

Given that the sum of two numbers

 x + y = 15 ..(i)

Given that the difference of two numbers

x - y = 3 ..(ii)

adding (i) and (ii) , we get

x + y + x -y = 15 +3

  2 x = 18

    x =9

Step(ii):-

Substitute x = 9 in equation (i) , we get

x + y = 15

9 + y = 15

     y = 15 - 9

    y = 6

Final answer:-

The numbers are  x= 9 and y =6

The product of two numbers

x y  = 9 × 6 = 54