Question
A red balloon is 40 feet above the ground and rising at 2ft/s . At the same time , a blue balloon is at 60 feet above the ground and descending at 3 ft/s , What will the height of the balloon be when they are the same height above the ground ?
Hint:
1. Speed =
2. New height = Current height (+/-) distance travelled by the balloon.
The correct answer is: Both the balloons will be at a height of 48 ft. when they are at the same height above the ground.
Step-by-step solution:-
Let the 2 balloons be b1 & b2, respectively and the time taken by them be x sec
As per the given diagram, we get-
Current height of b1 = 40 ft.
Speed of b1 = 2 ft/sec
Current height of b2 = 60 ft.
Speed of b2 = 3 ft/sec
We know that-
Distance covered by b1 upwards = current speed × time ............................. (Speed = )
∴ Distance covered by b1 upwards = 2 × x
∴ Distance covered by b1 upwards = 2x ................................... (Equation i)
New height of b1 = current height of b1 + distance covered upwards
∴ New height of b1 = 40 + 2x ...................................... (From Equations i & given information) ............................ (Equation ii)
Distance covered by b2 downwards = current speed × time ............................. (Speed = Distance/Time)
∴ Distance covered by b2 downwards = 3 × x
∴ Distance covered by b2 downwards = 3x ................................... (Equation iii)
New height of b2 = current height of b2 - distance covered downwards
∴ New height of b2 = 60 - 3x ....................................... (From Equations iii & given information) ........................... (Equation iv)
Now, we need to find the value of x for which the 2 balloons will be at the same height.
∴ New height of b1 = New height of b2
∴ 40 + 2x = 60 - 3x ….............................. (From Equations ii & iv)
∴ 2x + 3x = 60 - 40 ................................. (Taking variables & constants on either sides of the equation)
∴ 5x = 20
∴ x = 
∴ x = 4 sec
Height of b1 at x = 4 i.e. after 4 seconds = 40 + 2x
∴ Height of b1 at x = 4 i.e. after 4 seconds = 40 + 2 × 4
∴ Height of b1 at x = 4 i.e. after 4 seconds = 40 + 8
∴ Height of b1 at x = 4 i.e. after 4 seconds = 48 ft.
Height of b2 at x = 4 i.e. after 4 seconds = 60 - 3x
∴ Height of b2 at x = 4 i.e. after 4 seconds = 60 - 3 × 4
∴ Height of b2 at x = 4 i.e. after 4 seconds = 60 - 12
∴ Height of b2 at x = 4 i.e. after 4 seconds = 48 ft.
Final Answer:-
∴ Both the balloons will be at a height of 48 ft. when they are at the same height above the ground.
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