A hawk swoops from its perch 30 feet in a tree to catch a field mice. It follows the function h(t)=6t^2-24t+30 where t is time in seconds and h is height in feet

Question

What is the initial height of the hawk? Write as ordered pair
At which point is the hawk at its lowest height? Write as ordered pair
The hawk will be at its lowest point in time __ seconds at a height of ___ feet

Answer 1

To analyze the function given for the hawk's height, \( h(t) = 6t^2 - 24t + 30 \), we need to find several key points.

Initial Height

The initial height of the hawk (when \( t = 0 \)) can be calculated by substituting \( t = 0 \) into the function:

\[ h(0) = 6(0)^2 - 24(0) + 30 = 30 \]

Thus, the initial height of the hawk is:

\[ (0, 30) \]

Lowest Height

To find the lowest height (the vertex of the quadratic function), we can use the vertex formula for a parabola given by \( t = -\frac{b}{2a} \) where \( a \) and \( b \) are coefficients from the quadratic equation \( at^2 + bt + c \).

In this case, \( a = 6 \) and \( b = -24 \):

\[ t = -\frac{-24}{2 \times 6} = \frac{24}{12} = 2 \text{ seconds} \]

Now we substitute \( t = 2 \) back into the height function to find the minimum height:

\[ h(2) = 6(2)^2 - 24(2) + 30 \] \[ h(2) = 6(4) - 48 + 30 \] \[ h(2) = 24 - 48 + 30 = 6 \text{ feet} \]

Thus, the lowest height of the hawk occurs at:

\[ (2, 6) \]

Summary of Results

• The initial height of the hawk is \( (0, 30) \).
• The hawk reaches its lowest height of \( 6 \text{ feet} \) at \( t = 2 \) seconds, which is at the point \( (2, 6) \).

\[ \text{The hawk will be at its lowest point in time } 2 \text{ seconds at a height of } 6 \text{ feet.} \]