Rita kicks a football. Its height h, in metres, after t seconds is given by
h = —4.9t^2 + 22.54t + 1.1. To one decimal place, find the maximum height of the football and the time when it reached this height.
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To find the maximum height of the football, we need to determine the vertex of the quadratic equation representing the height of the football during its flight. The vertex form of a quadratic equation is given by:
h = a(t - h)^2 + k
Comparing this with the equation provided, h = -4.9t^2 + 22.54t + 1.1, we can identify a = -4.9, b = 22.54, and c = 1.1.
The x-coordinate of the vertex can be found using the formula:
t = -b / (2a)
Plugging in the values, we have:
t = -22.54 / (2 * -4.9)
t = 2.3
Therefore, the time when the football reached its maximum height is 2.3 seconds.
To find the maximum height, substitute this time value back into the equation:
h = -4.9(2.3)^2 + 22.54(2.3) + 1.1
Calculating this expression, we find:
h ≈ 26.4
Therefore, the maximum height of the football is approximately 26.4 meters.