The point-slope form of the equation for the elevator's movement is y - 33 = 7(x - 4), with an x-intercept of approximately 8.71 seconds and a y-intercept of 5 meters. The equation rewritten into slope-intercept form is y = 7x + 5.
To answer the student's question about the elevator's movement, we can use the two points provided: (4, 33) and (6, 47), where 'x' represents time in seconds and 'y' represents height in meters. First, we calculate the slope (m) of the line using the two points:
m = (y2 - y1) / (x2 - x1) = (47 - 33) / (6 - 4) = 14 / 2 = 7.
Now, we can write the point-slope form of the equation using one of the points, let's use (4, 33):
y - y1 = m(x - x1)
y - 33 = 7(x - 4)
To find the x-intercept, set y to 0 and solve for x:
0 - 33 = 7(x - 4)
x = 33 / 7 + 4
x ≈ 8.71
To find the y-intercept, set x to 0 in the slope-intercept form, which we will derive next:
y = mx + b
y = 7x + b (plug in y = 33, x = 4)
33 = 7(4) + b
33 = 28 + b
b = 5
So the slope-intercept form is:
y = 7x + 5
The y-intercept is 5 meters, meaning when t = 0, the elevator is already 5 meters above the ground.
The final answer in point-slope form: y - 33 = 7(x - 4). The x-intercept is approximately 8.71 seconds, and the y-intercept is 5 meters. The slope-intercept form is y = 7x + 5.