Question

the quadratic function ( h(x)=-16t^{2}+100x + 10 ) models the height above the ground ( h ), in feet, of a projectile ( x ) seconds after it had been launched vertically. if ( y = h(x) ) is graphed in the ( xy )-plane which of the following represents the real life meaning of the positive ( x )-intercept of the graph?
the initial height of the projectile
the maximum height of the projectile
the time at which the projectile reaches its maximum height
the time at which the projectile hits the ground

Explanation:

Step1: Analyze the meaning of x - intercept

The x - intercept of the graph of \(y = h(x)\) occurs when \(y=h(x)=0\). In the context of the height function \(h(x)=-16x^{2}+100x + 10\), when \(h(x) = 0\), the height of the projectile is \(0\) (i.e., it hits the ground).

Step2: Analyze other options

• The initial height of the projectile is when \(x = 0\). Substitute \(x = 0\) into \(h(x)\): \(h(0)=-16\times0^{2}+100\times0 + 10=10\) (not related to x - intercept).
• The time at which the projectile reaches its maximum height is given by \(x=-\frac{b}{2a}\) for the quadratic function \(y = ax^{2}+bx + c\) (here \(a=-16\), \(b = 100\), \(x=\frac{-100}{2\times(-16)}=\frac{100}{32}=\frac{25}{8}\), related to the vertex, not x - intercept).
• The maximum height of the projectile is found by substituting \(x = \frac{25}{8}\) into \(h(x)\) (related to the vertex, not x - intercept).

Answer:

The time at which the projectile hits the ground