Related rates - finding hypotenuse of triangle

[cmkluza]

Homework Statement

A plane flying horizontally at an altitude of 3 mi and a speed of 480 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 4 mi away from the station. (Round your answer to the nearest whole number.)

Homework Equations

$a^2=b^2+c^2$ where $a$ is the hypotenuse of a triangle.

The Attempt at a Solution

I started by relating the given variables as follows

• Altitude is a constant of 3 mi
  
• Horizontal distance of the plane will be $x$. We measure when $x=4$ miles.
  
• Distance from the plane to the radar station will be $y$. We measure when $y^2=4^2+3^2 \longrightarrow y=5$ miles.
• Change in horizontal distance will be $\frac{dx}{dt}$ We are given that this is 480 miles/hour.

We have $y^2=x^2+3^2$. Taking the derivative with respect to time gives $2y\times\frac{dy}{dt}=2x\times\frac{dx}{dt}+0$. Substituting in known values gives: $2(5)\times\frac{dy}{dt}=2(4)(480) \longrightarrow \frac{dy}{dt}=\frac{4\times480}{5}=384$. Yet this is not the answer.

Where am I going wrong? I've actually drawn out the triangle and variables, and I'm fairly stuck as to which part I'm messing up. Any insight will be appreciated!

 

[phinds]

How did "4 mi away from the station" become only the horizontal component of how far the plane is away from the station?

 

[cmkluza]

How did "4 mi away from the station" become only the horizontal component of how far the plane is away from the station?

Looks like I totally misread the question. I thought that the problem statement stated 4 to be the distance traveled in the x direction, not the distance between the plane and the station. That's cleared everything up.

Using the above information, the value for $x$ is $\sqrt{4^2-3^2}=\sqrt{7}$. Plugging the correct values into my above derivative gives the correct answer to this problem: $2(4)\times\frac{dy}{dt}=2(\sqrt{7})(480)\longrightarrow\frac{dy}{dt}=\frac{\sqrt{7}\times480}{4}\approx317$.

Thanks for pointing that out!