An explosion has occurred in Canada. Its blast radius was a regular circle that can be modelled by the equation x2 +y2=100, where x and y are measured in km. The blast destroyed everything within the circle.
a) Canada has 4 fire stations located at the following locations (-8,10), (11,3), (6,-7) & (-4,-6). Using algebra prove which of the stations were destroyed by the explosion.
b) fireworks factory is located at (-6,-8). If fire not put out within 15 minutes it will reach the storage room causing another explosion. Which of the surviving fire stations can get trucks to the factory in time? We assume the trucks travel at an average speed of 80km/h.
a) To determine which fire stations were destroyed by the explosion, we need to check if the coordinates of the fire stations satisfy the equation of the blast radius.
Let's check each fire station one by one:
1. Station 1: (-8,10)
Substituting x = -8 and y = 10 into the equation x^2 + y^2 = 100:
(-8)^2 + (10)^2 = 64 + 100 = 164 ≠ 100
The fire station at (-8,10) was not destroyed.
2. Station 2: (11,3)
Substituting x = 11 and y = 3 into the equation x^2 + y^2 = 100:
(11)^2 + (3)^2 = 121 + 9 = 130 ≠ 100
The fire station at (11,3) was not destroyed.
3. Station 3: (6,-7)
Substituting x = 6 and y = -7 into the equation x^2 + y^2 = 100:
(6)^2 + (-7)^2 = 36 + 49 = 85 ≠ 100
The fire station at (6,-7) was not destroyed.
4. Station 4: (-4,-6)
Substituting x = -4 and y = -6 into the equation x^2 + y^2 = 100:
(-4)^2 + (-6)^2 = 16 + 36 = 52 ≠ 100
The fire station at (-4,-6) was not destroyed.
Therefore, none of the fire stations were destroyed by the explosion.
b) To determine which surviving fire stations can reach the fireworks factory in time, we need to calculate the travel time for each station and compare it to the time limit of 15 minutes.
The distance between a fire station and the fireworks factory can be calculated using the distance formula:
Distance = √[(x2 - x1)^2 + (y2 - y1)^2]
Using the given coordinates of the fire stations and the factory:
1. Station 1: (-8,10)
Distance = √[(-8 - (-6))^2 + (10 - (-8))^2] = √[4 + 324] = √328 ≈ 18.14 km
Travel time = Distance / Speed = 18.14 km / 80 km/h ≈ 0.23 hours ≈ 14 minutes
The fire station at (-8,10) can reach the factory within the time limit.
2. Station 2: (11,3)
Distance = √[(11 - (-6))^2 + (3 - (-8))^2] = √[289 + 121] = √410 ≈ 20.25 km
Travel time = Distance / Speed = 20.25 km / 80 km/h ≈ 0.253 hours ≈ 15.2 minutes
The fire station at (11,3) can reach the factory within the time limit.
3. Station 3: (6,-7)
Distance = √[(6 - (-6))^2 + (-7 - (-8))^2] = √[144 + 1] = √145 ≈ 12.04 km
Travel time = Distance / Speed = 12.04 km / 80 km/h ≈ 0.15 hours ≈ 9 minutes
The fire station at (6,-7) can reach the factory within the time limit.
4. Station 4: (-4,-6)
Distance = √[(-4 - (-6))^2 + (-6 - (-8))^2] = √[4 + 4] = √8 ≈ 2.83 km
Travel time = Distance / Speed = 2.83 km / 80 km/h ≈ 0.035 hours ≈ 2.1 minutes
The fire station at (-4,-6) can reach the factory within the time limit.
Therefore, all of the surviving fire stations can get trucks to the factory in time.