Question

An owl swoops down from its perch in a tree at 30 feet to catch field mice. It follows the function h=6t2 −24t+30
, where t is the time in seconds and h is the height in feet. Which of the following best interprets the vertex?(1 point)
Responses

(2,6)
left parenthesis 2 comma 6 right parenthesis

(6,2)
left parenthesis 6 comma 2 right parenthesis

(0,30)
left parenthesis 0 comma 30 right parenthesis

(30,0)
left parenthesis 30 comma 0 right parenthesis

Answer 1

To interpret the vertex of the function \( h = 6t^2 - 24t + 30 \), we first need to find the vertex of the quadratic equation. The vertex of a parabola represented by the equation \( h = at^2 + bt + c \) can be calculated using the formula for the time \( t \) at the vertex, given by:

\[
t = -\frac{b}{2a}
\]

For our equation:

- \( a = 6 \)
- \( b = -24 \)

Now, substituting the values into the formula:

\[
t = -\frac{-24}{2 \cdot 6} = \frac{24}{12} = 2
\]

Now we will find the height \( h \) at \( t = 2 \) by substituting \( t = 2 \) back into the original equation:

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

Thus, the vertex of the parabola occurs at the point \( (2, 6) \).

In the context of the problem, this means that at \( t = 2 \) seconds, the owl is at its highest point of 6 feet above the ground. Therefore, the best interpretation of the vertex is:

**(2,6)**