Question

on the back!

1. hunter flies his kite at 45 feet high for about half an hour. as the wind picks up, he lowers the kite to 30 feet. he flies it like this for a while. when the wind dies down, he lets the kite ascend to 45 feet again. sketch a graph of the function in terms of the height of the kite over time.

Explanation:

Step1: Analyze the height changes over time

• Initially, the kite's height is 45 feet for about 0.5 hours (30 minutes). So, from \( t = 0 \) to \( t = 0.5 \) hours, the height \( h(t)=45 \) (constant).
• Then, the wind picks up, and the height lowers to 30 feet. Assume this lowering happens over a short time (let's say from \( t = 0.5 \) to \( t = 0.6 \) hours for simplicity, but the key is the change in height). So, from \( t = 0.5 \) to \( t = 0.6 \), the height decreases from 45 to 30.
• Next, the kite flies at 30 feet for a while (let's say from \( t = 0.6 \) to \( t = 1.1 \) hours, same duration as the first constant period for symmetry, but the duration can be adjusted as per the "for a while" description). So, \( h(t) = 30 \) here.
• Then, the wind dies, and the height ascends back to 45 feet (from \( t = 1.1 \) to \( t = 1.2 \) hours, short time), and then repeats (since it says "again").

Step2: Sketch the graph components

• First segment (0 to 0.5 hours): A horizontal line at \( y = 45 \) (height) on the \( t - h \) graph (x - axis: time \( t \), y - axis: height \( h \)).
• Second segment (0.5 to 0.6 hours): A line with a negative slope (decreasing) from \( (0.5, 45) \) to \( (0.6, 30) \).
• Third segment (0.6 to 1.1 hours): A horizontal line at \( y = 30 \).
• Fourth segment (1.1 to 1.2 hours): A line with a positive slope (increasing) from \( (1.1, 30) \) to \( (1.2, 45) \).
• Fifth segment (1.2 to 1.7 hours): A horizontal line at \( y = 45 \) (repeating the first segment), and the pattern continues.

Step3: Plot the points and draw the graph

• Mark the time on the x - axis (in hours) and height on the y - axis (in feet).
• Plot the horizontal lines for the constant height periods and the sloped lines for the height changes. The key features are the two constant height levels (45 and 30 feet) and the linear changes between them.

Answer:

The graph will have horizontal segments at \( h = 45 \) and \( h = 30 \) (with the duration of the horizontal segments being approximately equal for the first two constant periods, as per the "about half an hour" and "for a while" descriptions) and sloped segments (with negative slope when lowering and positive slope when ascending) connecting these horizontal segments. The x - axis represents time (in hours) and the y - axis represents the height of the kite (in feet). The graph repeats the pattern of 45 feet (constant) → lower to 30 feet (linear decrease) → 30 feet (constant) → ascend to 45 feet (linear increase) → 45 feet (constant), and so on.