Answer:
Yes, Samuel's Dart does land on the dartboard.
Step-by-step explanation:
The diameter of the dartboard = 18 inches
∴ The radius of the circle = 9 inches = 0.75 feet
The coordinate of the bull's-eye = 11 feet from the left wall, 5 feet 8 inches from the ground
∴ The coordinate of the bull's-eye = (11, 5.67)
The position of the dart Samuel throws = 11.5 feet from the left wall, 5.5 feet from the ground
∴ The coordinates of the position of the dart Samuel throws = (11.5, 5.5)
The equation for the area of a circle is given as follows;
(x - h)² + (y - k)² = r²
Where;
(h, k) = The coordinates of the center
r = The radius
Given that the bull's-eye is the center of the circle formed by the dartboard of radius 9 inches, we have;
(h, k) = The coordinates of the center
For the dartboard, we have;
(h, k) = The coordinates of the center = The coordinate of the bull's-eye = (11, 5.67)
h = 11, k = 5.67
∴ The equation of the circle formed by the dartboard = (x - 11)² + (y - 5.67)² = 0.75² = (3/4)² = 9/16
The square of the distance of the dart Michael throws from the bull's eye = (11.5 - 11)² + (5.5 - 5.[tex]\overline 6[/tex]7)² = 5/18
Therefore, given that the square of the distance of the dart Michael throws from the bull's eye < The square of the radius of the circle formed by the dart board and that the center of the, where the length of both the position of Samuel's dart and the radius of the circle formed by the dartboard are measured from the same point, (the center of the circle) we have;
The position of Samuel's dart is on the dart board
Therefore;
Yes, Samuel's Dart does land on the dartboard.
