Question

a ball is thrown into the air with an upward velocity of 40ft/s. its height h in feet after t seconds is given by the function :
h(t) = -16t² + 40t + 10
how many seconds does it take the ball to reach its maximum height?
1.25 seconds
2.5 seconds
2 seconds
1.4 seconds

Explanation:

Step1: Identify the function type

The height function \( h(t) = -16t^2 + 40t + 10 \) is a quadratic function in the form \( ax^2 + bx + c \), where \( a = -16 \), \( b = 40 \), \( c = 10 \). For a quadratic function, the vertex (which gives the maximum or minimum) occurs at \( t = -\frac{b}{2a} \) when \( a < 0 \) (since the parabola opens downward, the vertex is a maximum).

Step2: Calculate the time at vertex

Substitute \( a = -16 \) and \( b = 40 \) into the formula \( t = -\frac{b}{2a} \).

$$ t = -\frac{40}{2 \times (-16)} = -\frac{40}{-32} = \frac{40}{32} = 1.25 $$

Answer:

1.25 seconds