Two surveyors with two-way radios leave the same point at 9:00 A.M., one walking due south at 3 mi/hr and the other due west at 4 mi/hr. How long can they communicate with one another if each radio has a maximum range of 7 miles?
To determine how long the surveyors can communicate with each other, we need to find the time it takes for them to get 7 miles apart.
Let's assume they can communicate as long as they are within the range of each other's radios. They are initially at the same point, so the surveyor walking south will move perpendicular to the surveyor walking west, forming a right triangle.
The speed of the surveyor walking due south is 3 mi/hr, and the speed of the surveyor walking due west is 4 mi/hr. This means that their speeds form the sides of a right triangle with the time as the hypotenuse.
We can use the Pythagorean theorem to find the distance between them:
Distance^2 = (3*t)^2 + (4*t)^2,
Distance^2 = 9t^2 + 16t^2,
Distance^2 = 25t^2.
Since the maximum range of the radios is 7 miles, the distance between them must be 7 miles:
25t^2 = 7^2,
25t^2 = 49.
Dividing both sides by 25, we get:
t^2 = 49/25,
t^2 = 1.96.
Taking the square root of both sides, we get:
t = √1.96.
Therefore, the surveyors can communicate with each other for approximately √1.96 hours, which is approximately 1.4 hours or 1 hour and 24 minutes.
To figure out how long the surveyors can communicate with each other, we need to find out when they are at a distance of 7 miles from each other.
Let's assume the point they start at is the origin (0, 0) on a coordinate plane.
The surveyor walking due south at 3 mi/hr will have coordinates (0, -3t), and the surveyor walking due west at 4 mi/hr will have coordinates (-4t, 0), after t hours.
To find when they are 7 miles apart, we can use the distance formula:
d = √((x₂ - x₁)² + (y₂ - y₁)²)
where (x₁, y₁) and (x₂, y₂) are the coordinates of the two surveyors.
For the given problem, we have:
d = √((-4t - 0)² + (0 - (-3t))²)
= √(16t² + 9t²)
= √(25t²)
= 5t
We want to find the time (t) it takes for them to be 7 miles apart:
5t = 7
Divide both sides by 5:
t = 7/5
t = 1.4 hours
Therefore, the surveyors can communicate with each other for 1.4 hours.