Question

A hawk swoops down from its perch 30 feet in a tree to catch field mice. It follows the function h(t)=6t2−24t+30
, where t
is time in seconds and h
is height in feet.

Part A: Graph this function. Link to Desmos calculator

Part B: What is the initial height of the hawk? Write your answer as an ordered pair

Part C: At which point is the hawk at it's lowest height? Write your answer as an ordered pair

Part D: The hawk will be at it's lowest point at time
seconds at a height of
feet

Answer 1

Let's take your questions step by step.

### Part A: Graph the Function
You can graph the function \( h(t) = 6t^2 - 24t + 30 \) using the Desmos graphing calculator. Here is how you can set it up:

1. Go to the Desmos calculator website.
2. Enter the equation: `h(t) = 6t^2 - 24t + 30`.
3. Observe the parabola which opens upwards, as the coefficient of \( t^2 \) is positive.

### Part B: Initial Height of the Hawk
To find the initial height of the hawk, we evaluate \( h(t) \) at \( t = 0 \):

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

Therefore, the initial height of the hawk is 30 feet. The ordered pair for the initial height is:

**Ordered Pair:** \( (0, 30) \)

### Part C: Lowest Height of the Hawk
To find the lowest height of the hawk, we need to find the vertex of the parabola represented by the function \( h(t) = 6t^2 - 24t + 30 \). The \( t \)-coordinate of the vertex can be found using the formula:

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

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

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

Now we can find the height at \( t = 2 \):

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

Thus, the lowest height of the hawk is 6 feet. The ordered pair for this point is:

**Ordered Pair:** \( (2, 6) \)

### Part D: Summary of the Lowest Point
The hawk will be at its lowest point at:

- **Time:** \( 2 \) seconds
- **Height:** \( 6 \) feet

So you can summarize:

- The hawk will be at its lowest point at time **2 seconds** at a height of **6 feet**.