Rebecca is flying a drone at a constant height. She decides to make the drone rise vertically. It rises 18m every 3s. After 5s, the drone is at a height of 40m. The drone’s height in meters, y, is a function of the time in seconds, x.
The correct answer and explanation is:
We are given a problem in which Rebecca is flying a drone at a constant height and then decides to make it rise vertically. The drone’s height, yy, is a function of time, xx, and we are tasked with finding this relationship.
Step 1: Understand the given information
- The drone rises 18 meters every 3 seconds. This means the drone has a vertical speed of: Rate of rise=18 meters3 seconds=6 meters per second.\text{Rate of rise} = \frac{18 \text{ meters}}{3 \text{ seconds}} = 6 \text{ meters per second}.
- After 5 seconds, the drone’s height is 40 meters. This provides a specific point on the function: when x=5x = 5, y=40y = 40.
Step 2: Set up the general function
The drone’s height increases at a constant rate of 6 meters per second, so the function describing the height, yy, as a function of time, xx, will be linear. A linear function is of the form: y=mx+b,y = mx + b,
where mm is the slope (rate of change of height) and bb is the y-intercept (initial height when x=0x = 0).
From the information given:
- The slope m=6m = 6 meters per second (since the drone rises 6 meters every second).
- The equation for height becomes: y=6x+b.y = 6x + b.
Step 3: Use the point (5,40)(5, 40) to find bb
We are told that after 5 seconds, the height is 40 meters, i.e., when x=5x = 5, y=40y = 40. We can substitute these values into the equation to solve for bb: 40=6(5)+b.40 = 6(5) + b. 40=30+b.40 = 30 + b. b=40−30=10.b = 40 – 30 = 10.
Thus, the equation for the drone’s height as a function of time is: y=6x+10.y = 6x + 10.
Step 4: Interpretation
The function y=6x+10y = 6x + 10 tells us that:
- The drone’s height increases by 6 meters for every second that passes.
- When x=0x = 0 (at the start), the drone is at a height of 10 meters.
- After 5 seconds, the drone reaches a height of 40 meters, as given in the problem.
Conclusion
The height of the drone as a function of time is y=6x+10y = 6x + 10. This equation accurately models the drone’s vertical motion based on the given information.