When line y= x+4 and parabola y= 1/2x squared intersect at points A and B, find the length of the line segment AB
from Kumon level I math page 191b
To find the length of the line segment AB, we first need to find the coordinates of points A and B.
First, set the equations y = x + 4 and y = (1/2)x^2 equal to each other:
x + 4 = (1/2)x^2
Rearranging the equation, we get:
(1/2)x^2 - x - 4 = 0
Now, we can solve for x by using the quadratic formula:
x = [1 ± sqrt(1 + 4*(1/2)*4)] / 2(1/2)
x = [1 ± sqrt(9)] / 1
x = [1 ± 3] / 1
So, x can be either 4 or -2.
When x = 4:
y = 4 + 4
y = 8
Point A is (4, 8)
When x = -2:
y = -2 + 4
y = 2
Point B is (-2, 2)
Now, we can calculate the length of the line segment AB using the distance formula:
AB = sqrt((x2 - x1)^2 + (y2 - y1)^2)
AB = sqrt((4 - (-2))^2 + (8 - 2)^2)
AB = sqrt(6^2 + 6^2)
AB = sqrt(36 + 36)
AB = sqrt(72)
AB ≈ 8.49
Therefore, the length of the line segment AB is approximately 8.49.