Full Service?Problem:A communications tower is able to process mobile telephone calls only if the telephone is within a certain radius of the tower. Calls made from within the specified radius can be handled without any problems; the tower cannot process calls made from outside the radius; calls made on the border will receive intermittent service. (Note that in order to avoid problems intermittent service and no service areas, phone companies actually assume a hexagonal service area).The purpose of this activity is to determine all of the points on the border of the service area of a communications tower.

A.Assume that the tower is located at the origin, (0, 0). Find the distance between the tower and a mobile phone located at:

i) point (0, 5) ii) point (5, 0)   

iii) point (3, 4) iv) point (-4, 3).

d 0 0 2 5 0 2

d 0 2 52

d 5 units

d 5 0 2 0 0 2

d 52 0 2

d 5 units

d 3 0 2 4 0 2

d 32 42

d 5 units

d -4 0 2 3 0 2

d -4 2 32

d 5 units

ii) Can you suggest other points that would be 5 units away from the tower?

  

iii) If you were to connect all the possible positions for a mobile phone that is 5 units away from the tower, what shape would they trace out?

B. i) Plot the location of the tower and the various mobile phone locations from question 1 on the same grid.

(0, -5) (-5, 0) (3, -4)(4, -3) (-3, 4) (-4, 3)(-3, -4) (-4, -3)

Circle

C. i) What would be the distance between the tower (0, 0) and a mobile phone located at P (x, y)?

  

ii) If P is 5 units away from the tower, what equation could you form?

  

iii) What would the graph of your relationship look like? Explain.

d x 0 2 y 0 2

d x2 y 2

d2 x2 y 2

x2 y 2 52

x2 y 2 25

Circle

D.Suggest a general equation you could use to represent all circles on coordinate plan with centre (0, 0) and radius, r.

r

0

P(x, y)x2 + y2 = r2

E. Use the equation you made in part D to answer the following questions.

 1. State the equation for each circle below.a) radius 3 units, centre (0, 0):

b) radius 25 units, centre (0, 0):  2. State the centre and the radius of

each circle.a) x2 + y2 = 36 b) x2 + y2 = 52

x2 + y2 = 9

x2 + y2 = 625

centre: (0, 0) radius = 6

centre: (0, 0)radius = 52

3. State the x and y-intercepts for the circle, x2 + y2 = 16.

y

0 (4, 0)xr

radius is 4

(-4, 0)

(0, -4)

(0, 4)

r

rr

4. Determine the equation of the circle that has centre (0, 0) and passes through point (– 4, 2).

x

y

0

r

Name other points that lie on the circle.

(-4, 2) (4, 2)

(-4, -2) (4, -2)

(-2, 4) (2, 4)

(-2, -4) (2, -4)

x2 + y2 = r2

(-4)2 + (2)2 = r2

20 = r2

x2 + y2 = 20

5. Challenge: Would points such as A(-3, -7), B(5, -12) and C(11, 8) receive no service, full service, or intermittent service if the service radius around a tower is 13 units and the tower is located at the origin?

rA = 7.62 unitsThe point A lies inside the circle and will receive full service.

rA = (-3)2 + (-7)2

rB = 13 unitsThe point B lies on the circle and will receive intermittent service.

rB = (5)2 + (-12)2

rC = 13.6 unitsThe point C lies outside the circle and will receive no service.

rC = (11)2 + (8)2

6. Super Challenge: If the point (a, 5) lies on the circle, x2 + y2 = 29, determine the value of a.

x2 + y2 = 29 substitute x = a and y = 5a2 + 52 = 29

a2 = 29 – 25 a2 = 4 a = ±2

√√

(-2, 5) and (2, 5)

7. Super Challenge: A raindrop falls into a puddle and causes a circular ripple. The radius of the ripple grows at a steady rate of 4 cm/s. What equation would model the ripple 10 seconds after the raindrop lands in the puddle?Recall, distance = speed x time

distance = (4 cm/s)(10 s) distance = 40 cm

radius = 40 cm x2 + y2 = r2 x2 + y2 = 402 x2 + y2 = 1600

the equation that models the ripple 10 s after the raindrop lands in the puddle is x2 + y2 = 1600.