Two flares are launched from a life raft. The height h of the first flare in metres above the sea is modelled by the formula h1= t2+t+1meters,and the second flare is h2=4-t where t is the time
12 in seconds after the flare is launched. Determine
(i) Time taken and
(ii) Height when the two flares meet each other.
Hint: When two flares meet each other, their heights are the same (h1=h2).
To determine the time taken and height when the two flares meet each other, we need to solve the equation h1 = h2.
Given:
h1 = t^2 + t + 1
h2 = 4 - t
Setting the heights equal to each other, we have:
t^2 + t + 1 = 4 - t
Rearranging the equation, we get:
t^2 + 2t - 3 = 0
To solve this quadratic equation, we can use the quadratic formula:
t = (-b ± √(b^2 - 4ac)) / (2a)
In this case, a = 1, b = 2, and c = -3. Plugging them into the formula, we have:
t = (-2 ± √(2^2 - 4(1)(-3))) / (2(1))
t = (-2 ± √(4 + 12)) / 2
t = (-2 ± √16) / 2
t = (-2 ± 4) / 2
This gives us two possible values for t:
t1 = (-2 + 4) / 2 = 2 / 2 = 1 second
t2 = (-2 - 4) / 2 = -6 / 2 = -3 seconds
Since time cannot be negative in this context, we discard t2 = -3 seconds. Therefore, the time taken for the two flares to meet is 1 second.
To find the height when the two flares meet, we substitute the value of t into either h1 or h2.
Using h1:
h1 = t^2 + t + 1
h1 = (1)^2 + (1) + 1
h1 = 1 + 1 + 1
h1 = 3 meters
Using h2:
h2 = 4 - t
h2 = 4 - (1)
h2 = 4 - 1
h2 = 3 meters
Therefore, the height when the two flares meet is 3 meters.