Water is flowing from a major broken water main at the intersection of two streets. The resulting puddle of water is circular and the radius r of the puddle is given by the equation r = 5t feet, where t represents time in seconds elapsed since the the main broke.
(a) When the main broke, a runner was located 6 miles from the intersection. The runner continues toward the intersec- tion at the constant speed of 17 feet per second. When will the runner’s feet get wet?
(b) Suppose, instead, that when the main broke, the runner was 6 miles east, and 5000 feet north of the intersection. The runner runs due west at 17 feet per sec- ond. When will the runner’s feet get wet?
(a) when 5t + 17t = 6*5280
(b) when 5t = √(5000^2+(6*5280-17t)^2)
(a) Well, well, well, the poor runner! Trying to outrun the flow of water, huh? Let's calculate when their feet are going to get wet.
Since the runner is moving towards the intersection at a constant speed of 17 feet per second, we need to find the time it takes for the runner to cover the distance of 6 miles (which is 6 miles x 5280 feet per mile because we're in the USA). That's a total distance of 31680 feet.
Now, we know that the radius of the puddle, r, is given by the equation r = 5t. We can equate this to the distance the runner has traveled, 31680 feet, and solve for t:
5t = 31680
t = 31680 / 5
t = 6336 seconds
So it will take the poor runner a whopping 6336 seconds to reach the intersection. That's more than an hour! Better bring some dry socks and swimming goggles, my friend.
(b) Ah, now the runner's in a different spot! Let's see when their feet will get wet this time.
The runner starts 6 miles east, which is 6 miles x 5280 feet per mile = 31680 feet east of the intersection. They run due west at a constant speed of 17 feet per second.
Since they're running west, they're moving directly towards the intersection. So how long will it take them to cover the distance of 31680 feet?
Time = Distance / Speed
Time = 31680 feet / 17 feet per second
Time ≈ 1863.529 seconds
It will take approximately 1863.529 seconds for the runner's feet to get wet. So grab an umbrella and a snorkel, my friend, because you're in for a wet run!
To solve this problem, we need to find the time at which the runner's distance from the intersection is equal to the radius of the puddle.
(a) When the main broke, the runner was located 6 miles from the intersection. We need to convert this distance to feet. Since 1 mile equals 5280 feet, the runner was 6 * 5280 = 31680 feet from the intersection.
The runner continues towards the intersection at the constant speed of 17 feet per second. Let's denote the time in seconds the runner has been running as t. The runner's distance from the intersection after t seconds can be calculated using the formula:
Distance = Initial distance + Speed * Time
So, after t seconds, the runner will be at a distance of 31680 - 17t from the intersection.
The radius of the puddle at time t is given by r = 5t. We need to find the time at which the runner's distance from the intersection is equal to r.
31680 - 17t = 5t
Combining like terms, we get:
31680 = 22t
Dividing both sides by 22, we get:
t = 31680 / 22
t ≈ 1440 seconds
Therefore, the runner's feet will get wet after approximately 1440 seconds.
(b) In this case, the runner started 6 miles east and 5000 feet north of the intersection. Again, we convert this distance to feet. The runner was 6 * 5280 = 31680 feet east and 5000 feet north of the intersection.
The runner runs due west at a speed of 17 feet per second. Let's denote the time in seconds the runner has been running as t. The runner's distance from the intersection after t seconds can be calculated using the formula:
Distance = Initial distance + Speed * Time
So, after t seconds, the runner will be at a distance of 31680 - 17t east and 5000 feet north of the intersection.
The radius of the puddle at time t is given by r = 5t. We need to find the time at which the runner's distance from the intersection is equal to r.
Using the Pythagorean theorem, we can find the distance between the runner and the intersection:
Distance^2 = (31680 - 17t)^2 + (5000)^2
Simplifying this equation, we get:
(31680 - 17t)^2 + (5000)^2 = (5t)^2
Expanding and combining like terms, we get:
100t^2 - 226080t + 1004160 = 0
Using the quadratic formula, we can solve for t:
t = (-b ± √(b^2 - 4ac)) / (2a)
Plugging in the values, we get:
t = (-(-226080) ± √((-226080)^2 - 4 * 100 * 1004160)) / (2 * 100)
Simplifying further, we get:
t = (226080 ± √(226080^2 - 4 * 100 * 1004160)) / 200
Calculating this using a calculator, we get:
t ≈ 1102 seconds or t ≈ 2086 seconds
Therefore, the runner's feet will get wet after approximately 1102 seconds or 2086 seconds, depending on the direction the runner chooses.
To solve this problem, we need to find the time at which the distance between the runner and the center of the puddle becomes less than or equal to the radius of the puddle.
(a) The runner is initially 6 miles (6,000 feet) away from the intersection. The runner is moving towards the intersection at a constant speed of 17 feet per second. At time t, the distance the runner has covered is given by d = 17t.
The distance between the runner and the center of the circular puddle is given by the formula r = 5t. We need to find the time t when the distance d equals r.
Let's set up an equation: d = r.
Substitute the expressions for d and r: 17t = 5t.
Simplify the equation: 17t - 5t = 0.
Combine like terms: 12t = 0.
Divide both sides by 12 to solve for t: t = 0.
At time t = 0, the runner is located 6,000 feet away from the intersection. The runner's feet will get wet at time t = 0.
(b) In this case, the runner is initially 6 miles east (6,000 feet east) and 5,000 feet north of the intersection. The runner is moving due west at a constant speed of 17 feet per second. The distance the runner has covered in the west direction is still given by d = 17t.
The distance between the runner and the center of the circular puddle is given by the formula r = 5t. We need to find the time t when the distance d equals r.
Since the runner is moving west, the distance east of the intersection decreases, and it will change the equation by adding or subtracting the initial distance. Therefore, we need to subtract the initial east distance of 6,000 feet from the equation: distance = 17t - 6,000.
Set up an equation: 17t - 6,000 = 5t.
Simplify the equation: 17t - 5t - 6,000 = 0.
Combine like terms: 12t - 6,000 = 0.
Add 6,000 to both sides: 12t = 6,000.
Divide both sides by 12 to solve for t: t = 500.
At time t = 500 seconds, the runner's feet will get wet.