How can answer this question ,help please?
Information:
A ship travels at a constant speed of 25 kilometres per hour (Kph) in a straight line from port A located at position (x_A, y_A) =(-100, -100) to port B at (x_B ,y_B )=(300 ,100)
During its journey ,the ship passes two lighthouses ,L_1 and L_2, which are located at positions (0,0) and (200.0), respectively.
Question:
Complete the square gives the following result for d^2_2
d^2_1=200000((t-3/10)^2 +1/100).
Explain how d_1 varies as t increases from 0 to 1. Determine the shortest distance between the ship and lighthouse L_1, to the nearest kilometre.
please need help
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To answer this question, we need to understand the given information and use the equation provided to calculate the distance d₁ and determine how it varies with time. We also need to find the shortest distance between the ship and lighthouse L₁.
Let's break down the steps:
Step 1: Understand the given information:
- The ship is traveling from port A (-100, -100) to port B (300, 100) at a constant speed of 25 Kph.
- There are two lighthouses, L₁ at (0, 0) and L₂ at (200, 0).
- The equation given is d₁² = 200000((t - 3/10)² + 1/100), where t represents time.
Step 2: Calculate d₁ and determine how it varies with time:
- Plug in different values of t from 0 to 1 into the equation and calculate d₁ for each value.
- For example, when t = 0, d₁² = 200000((0 - 3/10)² + 1/100) = 200000((9/100) + 1/100) = 200000(10/100) = 20,000.
- Similarly, calculate d₁ for different values of t: 0.1, 0.2, 0.3, ..., 0.9, 1.
- Plot these values on a graph to observe how d₁ varies with time.
Step 3: Find the shortest distance between the ship and lighthouse L₁:
- To find the shortest distance, we need to determine the point on the ship's path that is closest to L₁.
- Calculate the distance between L₁ (0, 0) and the line connecting points A and B.
- Find the perpendicular distance from L₁ to the line to determine the shortest distance. Use the formula for the distance of a point P(x,y) from a line Ax + By + C = 0 given by: d = |Ax₀ + By₀ + C| / sqrt(A² + B²).
Finally, round the shortest distance to the nearest kilometer.
I hope this explanation helps you understand how to approach and solve the problem.