The interval in which the height is increasing is from t = 0 to t = 17.08 seconds
Since the path of a basketball after being thrown can be modeled by the following function, where x is the horizontal distance (in feet) from where the ball was thrown and y is the corresponding height (in feet).
y = −0.03x² + 1.025x + 4
Interval in which the height is increasing
The interval in which the height is increasing is when the derivative with respect to x, dy/dx ≥ 0
So, dy/dx ≥ 0
⇒ d(−0.03x² + 1.025x + 4)/dx ≥ 0
⇒ -0.06x + 1.025 ≥ 0
⇒ -0.06x ≥ -1.025
Dividing through by -0.06,
⇒ x ≤ -1.025/-0.06
⇒ x ≤ 17.08
Since the minimum value of x is 0 and x ≤ 17.08, this implies that the maximum value of x = 17.08.
So, the interval in which the height is increasing is from t = 0 to t = 17.08 seconds
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