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

Question

ℎ
=
6
𝑡
2

−
24
𝑡
+
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)
(
2
,
6
)
left parenthesis 2 comma 6 right parenthesis

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

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

(30,0)

Answer 1

To find the vertex of the quadratic function \( h = 6t^2 - 24t + 30 \), we can use the vertex formula for a quadratic function of the form \( h(t) = at^2 + bt + c \). The vertex \( t \)-coordinate can be found using the formula:

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

Here, \( a = 6 \) and \( b = -24 \).

Calculating \( t \):

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

Next, we can find the corresponding height \( h \) at \( t = 2 \):

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

So the vertex of the function is \( (2, 6) \).

In this context, the vertex represents the maximum height the owl can reach during the hunt before descending. The height is 6 feet at 2 seconds.

Therefore, the correct interpretation of the vertex is:

\((2, 6)\).