How do I do this related rates problem?
A plane flying horizontally at an altitude of 1 mi and a speed of 510 mi/h passes directly over a radar station. Find the rate at which the distance from the plane to the station is increasing when it is 6 mi away from the station. (Round to the nearest whole number.)
D^2 = 1^2 + x^2
2D(dD/dt) = 2x(dx/dt)
when D = 6
36 = 1 + x^2
x=√35
dD/dt = 2(6)510)/(2(√35))
= 517.23 mi/h
this is wrong you solved for the wrong side you need to solve for the hypotenuse. 6 is not the hypotenuse its a side. so the actual is sqrt(37).
To solve this related rates problem, follow these steps:
Step 1: Understand the problem
We are given that a plane is flying horizontally at an altitude of 1 mi and a speed of 510 mi/h. The plane passes directly over a radar station, and we need to find the rate at which the distance from the plane to the station is increasing when it is 6 mi away from the station.
Step 2: Identify the relevant variables
Let's denote the distance from the plane to the radar station as "d" and the time as "t". We are given the rate of change of the distance, which is dx/dt = 510 mi/h. We are asked to find the rate at which the distance is changing, which is d(d/dt) when d = 6 mi.
Step 3: Set up the equation
To relate the variables, we can use the Pythagorean theorem:
d^2 = altitude^2 + distance^2
Substituting the given values, we have:
d^2 = 1^2 + distance^2
Step 4: Differentiate both sides of the equation with respect to time
Differentiating both sides of the equation implicitly with respect to time (t), we get:
2d * (d/dt) = 0 + 2(distance * d(distance)/dt)
Simplifying the equation, we have:
2d * (d/dt) = 2(distance * d(distance)/dt)
Step 5: Plug in the known values and solve for the unknown rate
We are given that d = 6 mi and dx/dt = 510 mi/h. Substituting these values into the equation, we get:
2 * 6 * (d/dt) = 2(distance * d(distance)/dt)
12 * (d/dt) = 2(distance * d(distance)/dt)
Simplifying further:
12 * (d/dt) = 2(distance * d(distance)/dt)
(d/dt) = (distance * d(distance)/dt) / 6
(d/dt) = (distance * d(distance)/dt) / 6
Now, we need to find the value of d(distance)/dt. Let's recall that dx/dt = 510 mi/h and d = 6 mi. We can find this rate by rearranging the Pythagorean theorem:
d^2 = (distance)^2
2d * (d/dt) = 2(distance) * (d(distance)/dt)
Simplifying further:
12 * (d/dt) = 2(distance) * (d(distance)/dt)
Next, plug in d = 6 and dx/dt = 510, and solve for d(distance)/dt:
12 * (510) = 2(6) * (d(distance)/dt)
6120 = 12 * (d(distance)/dt)
(d(distance)/dt) = 6120 / 12
(d(distance)/dt) = 510
So, the rate at which the distance from the plane to the station is increasing when it is 6 mi away from the station is 510 mi/h.
To solve this related rates problem, we need to use the concept of derivatives and related rates formulas. Follow these steps to find the rate at which the distance from the plane to the station is increasing:
1. Understand the problem: Read the problem carefully and identify the given information and what needs to be found. In this case, the given information is the altitude of the plane (1 mi) and its speed (510 mi/h), and we need to find the rate at which the distance from the plane to the station is increasing.
2. Draw a diagram: Visualize the scenario by drawing a diagram. Draw a horizontal line representing the ground, and label one point as the radar station. Then, mark another point above the ground to represent the plane flying horizontally at an altitude of 1 mi.
3. Find the relevant equation: Identify the equation that relates the given information and what needs to be found. In this case, we can use the Pythagorean theorem to relate the distance from the plane to the station with the altitude of the plane.
The Pythagorean theorem states that for any right triangle, the square of the length of the hypotenuse (c) is equal to the sum of the squares of the other two sides (a and b).
In our case, the hypotenuse represents the distance from the plane to the station, and the other two sides represent the altitude of the plane (1 mi) and the distance traveled by the plane (which is what we need to find).
Therefore, the equation is: c^2 = a^2 + b^2, or d^2 = h^2 + x^2, where d is the distance from the plane to the station, h is the altitude of the plane (1 mi), and x is the distance traveled by the plane (which is what we need to find).
4. Take derivatives: Differentiate both sides of the equation with respect to time (t) to find the rates of change. Since we are looking for the rate at which the distance (d) is changing with respect to time (t), we need to differentiate both sides with respect to time.
d^2 = h^2 + x^2
Differentiating both sides with respect to t:
2d * dd/dt = 2h * dh/dt + 2x * dx/dt
5. Plug in the known values: Substitute the given values into the equation. The altitude of the plane (h) is 1 mi, and we are given that the speed of the plane (dx/dt) is 510 mi/h. We need to find the distance from the plane to the station (d) when it is 6 mi away (x = 6 mi).
2d * dd/dt = 2(1) * (dh/dt) + 2(6) * (510)
6. Solve for the unknown rate: Solve the equation for the unknown rate, which is dd/dt. Rearrange the equation to isolate dd/dt on one side, and then solve for it.
dd/dt = (12 * 510) / (2d)
dd/dt = 6120 / d
7. Substitute the value and solve: Plug in the value of d (6 mi since the plane is 6 mi away from the station) into the equation and calculate the result.
dd/dt = 6120 / 6
dd/dt = 1020 mi/h
Therefore, the rate at which the distance from the plane to the station is increasing when it is 6 mi away from the station is approximately 1020 mi/h, rounded to the nearest whole number.