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.