The cup on the9th hole of a golf course is located dead center in the middle of a circular green that is70 feet in diameter. Your ball is located as in the picture below:
The ball follows a straight line path and exits the green at the right-most edge. Assume the ball travels a constant rate of10 ft/sec.
(a) Where does the ball enter the green?
(b) When does the ball enter the green?
(c) How long does the ball spend inside the green?
(d) Where is the ball located when it is closest to the cup and when does this occur.
We do not have the picture.
To find the answers to the given questions, we can break down the problem into smaller parts and use some formulas from geometry and physics.
(a) To determine where the ball enters the green, we need to determine the point where its straight line path intersects with the circular green. Since the ball exits at the right-most edge, we know that the point of entry will be on the left side of the green.
To find this point, we can use the equation of a circle: (x - h)^2 + (y - k)^2 = r^2. Here, (h, k) represents the center of the circle, and r represents the radius.
In this case, the center of the circle is the dead center of the green, so (h, k) = (0, 0). The radius is half the diameter, so r = 70/2 = 35 feet.
Using the equation of the circle, we can substitute the values and solve for x:
(x - 0)^2 + (y - 0)^2 = 35^2
x^2 + y^2 = 1225
However, we also know that the x-coordinate of the point of entry will be negative since it is on the left side. Therefore, we can simplify the equation:
x^2 + y^2 = 1225
x^2 = 1225 - y^2
x = -sqrt(1225 - y^2)
This equation gives us the x-coordinate of the point of entry in terms of y.
(b) To find when the ball enters the green, we need to determine the time it takes for the ball to travel the distance from its initial position to the point of entry.
The initial position of the ball is not provided in the question, so we cannot directly find the time. However, we can use the distance formula and the given constant rate of 10 ft/sec to find the time indirectly.
The distance between the ball's initial position and the point of entry is the magnitude of the x-coordinate of the point of entry, which is |-sqrt(1225 - y^2)| = sqrt(1225 - y^2).
Using the formula distance = rate * time, we can set up the following equation:
sqrt(1225 - y^2) = 10t
Here, t represents the time in seconds.
(c) To find how long the ball spends inside the green, we need to determine the time it takes for the ball to travel through the green. Since the exit point is not specified, we will assume that the ball exits the green at the right-most edge. Therefore, the ball will travel across the diameter of the green.
The diameter of the green is given as 70 feet. Using the formula distance = rate * time, we can set up the following equation:
70 = 10t
Solving for t, we find t = 7 seconds.
Therefore, the ball spends 7 seconds inside the green.
(d) To find where the ball is located when it is closest to the cup, we need to determine the point on the ball's path that is closest to the center of the cup. This point will be the closest distance from the cup.
Since the cup is located dead center in the middle of the circular green, the center coordinates of the cup are the same as the center coordinates of the green: (0, 0).
To find the point on the ball's path that is closest to the cup, we need to find the perpendicular distance from the center of the cup to the line of the ball's path.
The equation of the line can be represented as y = mx + b, where m represents the slope of the line and b represents the y-intercept. In this case, the line is given as a straight line so the slope is constant.
Using the given information that the ball travels in a straight line path and exits the green at the right-most edge, we can determine that the equation of the line is y = 0 (since it exits at the right-most edge).
The perpendicular distance from the center of the cup to the line y = 0 is simply the y-coordinate of the center of the cup.
Therefore, the ball is closest to the cup when it is located at (0, 0), which is the center of the cup.
To find when this occurs, we can refer back to part (b) and find the time it takes for the ball to reach the center of the cup.