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maksim [4K]
2 years ago
10

A radio telescope has a parabolic surface, as shown below.

Mathematics
1 answer:
krek1111 [17]2 years ago
5 0
OK, so the graph is a parabola, with points x=0,y=0; x=6,y=-9; and x=12,y=0

Because the roots of the equation are 0 and 12, we know the formula is therefore of the form

y = ax(x - 12), for some a

So put in x = 6

-9 = 6a(-6)

9 = 36a

a = 1/4

So the parabola has a curve y = x(x-12) / 4, which can also be written y = 0.25x² - 3x

The gradient of this is dy/dx = 0.5x - 3

The key property of a parabolic dish is that it focuses radio waves travelling parallel to the y axis to a single point. So we should arrive at the same focal point no matter what point we chose to look at. So we can pick any point we like - e.g. the point x = 4, y = -8

Gradient of the parabolic mirror at x = 4 is -1

So the gradient of the normal to the mirror at x = 4 is therefore 1.

Radio waves initially travelling vertically downwards are reflected about the normal - which has a gradient of 1, so they're reflected so that they are travelling horizontally. So they arrive parallel to the y axis, and leave parallel to the x axis.

So the focal point is at y = -8, i.e. 1 metre above the back of the dish.
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Answer:

x=2 or x=−6

Step-by-step explanation:

4 0
3 years ago
Do you know the answer toPart A: 25% of 50 = Part B: 25% of 60 = Part C: 50% of 60 = Part D: 75% of 60 = Part E: 75% of 30 = Par
Verdich [7]

Answer:

Part A: 25% of 50 = 12.5

Part B: 25% of 60 = 15

Part C: 50% of 60 = 30

Part D: 75% of 60 = 45

Part E: 75% of 30 = 22.5

Part F: 100% of 22.5 = 22.5

Part G: 10% of 22.5 = 2.25

Part H: 50% of 45.7 = 22.85

Step-by-step explanation:

To find - Do you know the answer to

              Part A: 25% of 50 =

              Part B: 25% of 60 =

              Part C: 50% of 60 =

              Part D: 75% of 60 =

              Part E: 75% of 30 =

              Part F: 100% of 22.5 =

              Part G: 10% of 22.5 =

              Part H: 50% of 45.7 =

Proof -

Part A :

25% of 50 =  \frac{25}{100} *50 = \frac{1}{4}*50 = 12.5

⇒25% of 50 = 12.5

Part B :

25% of 60 =  \frac{25}{100} *60 = \frac{1}{4}*60 = 15

⇒25% of 60 = 15

Part C :

50% of 60 =  \frac{50}{100} *60 = \frac{1}{2}*60 = 30

⇒50% of 60 = 30

Part D :

75% of 60 = \frac{75}{100} *60 = \frac{75}{10}*6 = 45

⇒75% of 60 = 45

Part E :

75% of 30 = \frac{75}{100} *30 = \frac{75}{10}*3 = 22.5

⇒75% of 30 = 22.5

Part F :

100% of 22.5 = \frac{100}{100} *22.5 = 22.5

⇒ 100% of 22.5 = 22.5

Part G :

10% of 22.5 = \frac{10}{100} *22.5 = 2.25

⇒ 10% of 22.5 = 2.25

Part H :

50% of 45.7 =  \frac{50}{100} *45.7 = \frac{1}{2}*45.7 = 22.85

⇒50% of 45.7 = 22.85

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