A stranded soldier shoots a signal flare into the air to attract the attention of a nearby plane. The flare has an initial vertical velocity of 1,500 feet per second. Its height is defined by the quadratic function below. Assume that the flare is fired from ground level. \[ h=v_{i} t-16 t^{2} \] 10. What is the maximum height that the flare reaches? Show your work.

To find the maximum height that the flare reaches, we can use the given quadratic function for height: \[ h(t) = v_i t - 16 t^2 \] where \( v_i = 1500 \) feet per second. Thus, the function becomes: \[ h(t) = 1500t - 16t^2 \] This is a quadratic function in the standard form \( h(t) = at^2 + bt + c \), where: - \( a = -16 \) - \( b = 1500 \) - \( c = 0 \) The maximum height of a quadratic function occurs at the vertex, which can be found using the formula for the time \( t \) at which the maximum height occurs: \[ t = -\frac{b}{2a} \] Substituting in our values for \( a \) and \( b \): \[ t = -\frac{1500}{2 \times -16} = \frac{1500}{32} = 46.875 \text{ seconds} \] Now, we can substitute this value of \( t \) back into the height function to find the maximum height: \[ h(46.875) = 1500(46.875) - 16(46.875)^2 \] Calculating \( 1500(46.875) \): \[ 1500 \times 46.875 = 70312.5 \] Now calculating \( 16(46.875)^2 \): \[ (46.875)^2 = 2190.140625 \] \[ 16 \times 2190.140625 = 35062.25 \] Now substituting these values back into the height equation: \[ h(46.875) = 70312.5 - 35062.25 = 35250.25 \] Thus, the maximum height that the flare reaches is: \[ \boxed{35250.25} \text{ feet} \]