Question

lesson 6.1: absolute value and absolute value functions

1. a hot air balloon pilot raises a balloon 200 m from the ground, turns off the burner, and allows the balloon to descend 60 m before turning the burner on again to raise it another 170 m.

a. use positive and negative values to represent each stage of the balloon’s flight path.
b. use absolute values to determine the total vertical distance travelled by the balloon.

1. during a volleyball game, christine dove to save a ball. the function $h(t) = -4.9t^2 + 8.5t$ approximates the height of the ball above the ground in metres, $t$ seconds after it is hit.

a. if the net is 2.23 m tall, the function $d(t) = |-4.9t^2 + 8.5t - 2.23|$ represents the distance between the bottom of the ball and the top of the net after $t$ seconds. explain this function.

Explanation:

Part 6a

Step1: Define positive/negative for motion

Let upward motion be positive, downward be negative.

• Raising 200 m: \( +200 \, \text{m} \)
• Descending 60 m: \( -60 \, \text{m} \)
• Raising 170 m: \( +170 \, \text{m} \)

Part 6b

Step1: Recall absolute value for distance

Total vertical distance is sum of absolute values of each displacement.

Step2: Calculate each absolute value

\( |+200| + |-60| + |+170| = 200 + 60 + 170 \)

Step3: Sum the values

\( 200 + 60 + 170 = 430 \)

Part 7a

The function \( h(t) = -4.9t^2 + 8.5t \) gives the ball’s height. The net is 2.23 m tall. The distance between the ball’s bottom (height \( h(t) \)) and the net’s top (2.23 m) is \( |h(t) - 2.23| \), since distance is non - negative. Substituting \( h(t) \), we get \( d(t)=|-4.9t^2 + 8.5t - 2.23| \), which represents this vertical distance at time \( t \).

Answer:

s:
6a. Raising 200 m: \( +200 \, \text{m} \); Descending 60 m: \( -60 \, \text{m} \); Raising 170 m: \( +170 \, \text{m} \)
6b. \( 430 \, \text{metres} \)
7a. The function \( d(t)=|-4.9t^2 + 8.5t - 2.23| \) is derived from the vertical distance formula (distance is absolute difference) between the ball’s height (\( h(t) \)) and the net’s height (2.23 m), ensuring a non - negative distance at time \( t \).