Juliet and Mercutio are mov- ing at constant speeds in the xy-plane. They start moving at the same time. Juliet starts at the point (0, − 6) and heads in a straight line toward the point (10,5), reaching it in 10 sec- onds. Mercutio starts at (9, − 14) and moves in a straight line. Mercutio passes through the same point on the x axis as Juliet, but 2 sec- onds after she does.
How long does it take Mercutio to reach the y-axis?
To find out how long it takes Mercutio to reach the y-axis, we first need to determine the coordinates of the point on the x-axis that both Juliet and Mercutio pass through.
The slope of the line connecting Juliet's starting point (0, -6) and her destination (10, 5) can be found by calculating the vertical change divided by the horizontal change:
slope = (5 - (-6)) / (10 - 0)
= 11 / 10
Since Mercutio passes through the same point on the x-axis as Juliet, the line connecting Mercutio's starting point (9, -14) and the x-axis point will also have the same slope.
Using the slope-intercept form of a linear equation (y = mx + b), where m is the slope and b is the y-intercept, we can calculate the equation for the line passing through Juliet's point:
-6 = (11/10)(0) + b
Simplifying the equation, we find that b = -6.
Now, we can write the equation for the line that Mercutio follows:
y = (11/10)x + (-6)
To find the x-coordinate where Mercutio crosses the y-axis, we set y to 0 and solve for x:
0 = (11/10)x - 6
(11/10)x = 6
x = (10/11)(6)
x ≈ 5.45
Mercutio crosses the y-axis at the point (5.45, 0). To determine how long it takes for Mercutio to reach the y-axis, we need to measure the time elapsed from when Juliet started moving.
Juliet reaches the x-coordinate 5.45 in 10 seconds, so Mercutio would reach the same point 2 seconds later, with a total time of:
10 seconds + 2 seconds = 12 seconds
Hence, it takes Mercutio 12 seconds to reach the y-axis.
To find out how long it takes for Mercutio to reach the y-axis, we need to determine the point where Mercutio intersects the x-axis.
First, let's find the equation of the line that Juliet is moving along. We can use the two given points (0, -6) and (10, 5) to find the slope (m) of the line. The slope is given by:
m = (y2 - y1) / (x2 - x1)
= (5 - (-6)) / (10 - 0)
= 11 / 10
Using the point-slope form of a linear equation, we can write the equation of the line that Juliet is moving along as:
y - y1 = m(x - x1)
y - (-6) = (11/10)(x - 0)
y + 6 = (11/10)x
Now, let's find the point on the x-axis that Mercutio intersects. We know that Mercutio passes through the same point as Juliet but 2 seconds later. Since Juliet takes 10 seconds to reach the point (10, 5), Mercutio reaches the same x-coordinate in 8 seconds.
Using the equation we found for Juliet's path, let's substitute the x-coordinate of the intersection point, which is the same as Juliet's intersection point, into the equation:
y + 6 = (11/10)(10 - 8)
y + 6 = (11/10)(2)
y + 6 = 22/10
y = 22/10 - 6
y = -38/10
y = -19/5
So, the point where Mercutio intersects the x-axis is (2, 0).
Now that we have the coordinates of the intersection point (2, 0), we can find the equation of the line that Mercutio is moving along. We'll use the two given points (9, -14) and (2, 0) to find the slope (m) of the line. The slope is given by:
m = (y2 - y1) / (x2 - x1)
= (0 - (-14)) / (2 - 9)
= 14 / (-7)
= -2
Using the point-slope form of a linear equation, we can write the equation of the line that Mercutio is moving along as:
y - y1 = m(x - x1)
y - 0 = (-2)(x - 2)
y = -2x + 4
To find out how long it takes for Mercutio to reach the y-axis, we need to find the x-coordinate when y equals zero in the equation we found for Mercutio's path:
0 = -2x + 4
2x = 4
x = 2
So, Mercutio reaches the y-axis at x = 2. Since he started at x = 9 and took 8 seconds to reach the point Juliet crossed, the time it takes for Mercutio to reach the y-axis would be 8 seconds.