Question

1. the figure below shows the parabolic path for a scuba divers dive off of a boat.

a) what is initial height of the scuba diver?
b) what is the minimum height the diver?
c) how far away was the diver from the boat when he resurfaced?

Explanation:

Part (a)

Step 1: Identify initial point

The initial height occurs at \( x = 0 \) (distance from boat is 0). Check the y - value at \( x = 0 \) on the graph.
From the graph, when \( x = 0 \), the \( y \) - coordinate (height) is 0? Wait, no, looking at the graph, the y - axis: the initial point (when \( x = 0 \)) is at \( y = 0 \)? Wait, no, the y - axis is labeled "Scuba Diver's Height (in feet)". Wait, the graph starts at \( x = 0 \), and the y - value at \( x = 0 \): looking at the graph, the point at \( x = 0 \) is at \( y = 0 \)? Wait, no, maybe I misread. Wait, the y - axis: the origin is (0,0)? Wait, the graph shows the diver's path. At \( x = 0 \) (distance from boat is 0), the height is 0? Wait, no, maybe the initial height is when \( x = 0 \), so we look at the y - intercept. From the graph, when \( x = 0 \), the \( y \) - value is 0? Wait, no, the graph has a point at \( x = 0 \), \( y = 0 \)? Wait, maybe I made a mistake. Wait, the y - axis: the labels are 15, 10, 5, 0, - 5, - 10, etc. At \( x = 0 \), the point is at \( y = 0 \)? Wait, no, the graph's y - axis: the initial position (when the diver starts, \( x = 0 \)) is at \( y = 0 \)? Wait, maybe the initial height is 0 feet? Wait, no, maybe I misread. Wait, the graph: the y - axis is "Scuba Diver's Height (in feet)". So when \( x = 0 \) (distance from boat is 0), the height is 0 feet.

Step 2: Confirm

So the initial height (at \( x = 0 \)) is 0 feet? Wait, no, maybe the graph's y - intercept: looking at the graph, when \( x = 0 \), the \( y \) - coordinate is 0. So the initial height is 0 feet.

Part (b)

Step 1: Identify minimum point

The minimum height of a parabola (since the path is parabolic) is at the vertex (the lowest point of the parabola).

Step 2: Read the y - value of the vertex

From the graph, the lowest point (vertex) has a y - value of - 30 feet? Wait, the y - axis labels: - 30 is one of the marks. So the minimum height is - 30 feet (or 30 feet below the surface, but height is measured as per the graph, so the minimum height is - 30 feet? Wait, the graph shows the vertex at \( y=-30 \), so the minimum height is - 30 feet (or 30 feet below the surface, but the height value is - 30 feet).

Part (c)

Step 1: Identify resurfacing point

Resurfacing occurs when the height \( y = 0 \) (since the surface is at \( y = 0 \)). We need to find the \( x \) - value when \( y = 0 \) (other than \( x = 0 \)).

Step 2: Find the x - intercept

From the graph, the parabola intersects the x - axis ( \( y = 0 \)) at \( x = 0 \) and \( x = 40 \). Since \( x = 0 \) is the starting point, the resurfacing occurs at \( x = 40 \) feet. So the distance from the boat is 40 feet.

Answer:

a) The initial height of the scuba diver is \(\boldsymbol{0}\) feet.
b) The minimum height of the diver is \(\boldsymbol{- 30}\) feet (or 30 feet below the surface).
c) The diver was \(\boldsymbol{40}\) feet away from the boat when he resurfaced.