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



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Answer 1

To solve the problem presented, we need to analyze the function \( h(t) = 6t^2 - 24t + 30 \) step by step.

### Part A: Graph this function
To graph the function, you can use the Desmos calculator. You would input the equation \( h(t) = 6t^2 - 24t + 30 \) into the graphing tool, and it will display the parabola that represents the height of the hawk over time.

### Part B: What is the initial height of the hawk?
The initial height occurs when \( t = 0 \). To find the initial height, we substitute \( t = 0 \) into the function:

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

So, the initial height of the hawk is 30 feet. As an ordered pair, the initial height is:

\[
(0, 30)
\]

### Part C: At which point is the hawk at its lowest height?
To find the lowest point (the vertex of the parabola), we can use the vertex formula for a quadratic function, which is given by \( t = -\frac{b}{2a} \), where \( a \) and \( b \) are the coefficients from the function \( h(t) = at^2 + bt + c \).

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

Calculating \( t \):

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

Now, substitute \( t = 2 \) back into the height function to determine the height at this time:

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

Thus, the lowest point of the hawk's flight occurs at:

\[
(2, 6)
\]

### Part D: The hawk will be at its lowest point at time __________ seconds at a height of __________ feet.
As we calculated in Part C, the hawk will be at its lowest point at:

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

So, you can fill in the blanks as:

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