a) The north boundary of the sidewalk is a horizontal line y = -90.
The south boundary of the sidewalk is a horizontal line y = -110.
b) The water will first strike the sidewalk after 2400 seconds.
c) The water from the sprinkler will fall completely north of the sidewalk after 3600 seconds.
d) The total amount of time water from the sprinkler falls on the sidewalk is 1200 seconds.
e) The area of grass watered after one hour is 400 pi square feet.
(a) The tractor will have x and y coordinates.
Initially, the tractor is 100 feet south of the sidewalk, so its y-coordinate is y = -100.
The tractor is on the hose, which is perpendicular to the sidewalk, which means its x-coordinate is constant.
Let's set the x-coordinate to be x = 0.
The north boundary of the sidewalk is a horizontal line y = -90.
The south boundary of the sidewalk is a horizontal line y = -110.
(b) The water from the sprinkler will first strike the sidewalk when the edge of the sprinkler's circle is on the sidewalk. This happens when the sprinkler is 20 feet north of the sidewalk, which means y = -80.
The time it takes for the sprinkler to move 20 feet north is:
time = distance / rate
time = 20 ft / (1/2 inch/sec) * (12 inches/ft) * (1/60 sec/min)
time = 2400 seconds
So, the water will first strike the sidewalk after 2400 seconds.
(c) The water from the sprinkler will fall completely north of the sidewalk when the center of the sprinkler's circle is north of the sidewalk. This happens when the sprinkler is 30 feet north of the sidewalk, which means y = -70.
The time it takes for the sprinkler to move 30 feet north is:
time = distance / rate
time = 30 ft / (1/2 inch/sec) * (12 inches/ft) * (1/60 sec/min)
time = 3600 seconds
So, the water from the sprinkler will fall completely north of the sidewalk after 3600 seconds.
(d) The total amount of time water from the sprinkler falls on the sidewalk is the difference between the time it takes for the water to first strike the sidewalk and the time it takes for the water to fall completely north of the sidewalk. This is:
total time = 3600 seconds - 2400 seconds
total time = 1200 seconds
So, the total amount of time water from the sprinkler falls on the sidewalk is 1200 seconds.
(e) The area of grass watered after one hour is the area of a circle with radius 20 feet that has been watered for one hour.
The area of a circle is:
area = pi * radius^2
area = pi * (20 ft)^2
area = 400 pi square feet
One hour is 60 minutes, so the area of grass watered after one hour is:
area = 60 minutes * 400 pi square feet / 60 minutes
area = 400 pi square feet
So, the area of grass watered after one hour is 400 pi square feet.