Questions: A plane flying horizontally at an altitude of 2 miles and a speed of 530 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 has a total distance of 5 miles away answer to the nearest whole number.

A plane flying horizontally at an altitude of 2 miles and a speed of 530 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 has a total distance of 5 miles away answer to the nearest whole number.

Transcript text: A plane flying horizontally at an altitude of 2 miles and a speed of $530 \mathrm{mi} / \mathrm{h}$ passes directly over a radar station. Find the rate at which the distance from the plane to the station is increasing when it has a total distance of 5 miles away answer to the nearest whole number.

Solution

Solution Steps

Step 1: Set up the relationship using the Pythagorean theorem

Given the altitude $h = 2$ miles and the horizontal distance $x$ from the radar station, the total distance $d$ from the plane to the radar station can be represented as $d^2 = x^2 + h^2$.

Step 2: Differentiate with respect to time $t$

Differentiating both sides of the equation with respect to $t$, we get $2d\frac{dd}{dt} = 2x\frac{dx}{dt} + 0$ since $h$ is constant and its derivative with respect to time is zero.

Step 3: Solve for $\frac{dd}{dt}$

Simplifying the differentiated equation, we find $\frac{dd}{dt} = \frac{x}{d}\frac{dx}{dt}$. Here, $\frac{dx}{dt}$ is the speed of the plane $v = 5$ miles per hour, which is given.

Step 4: Substitute known values and solve

Using the Pythagorean theorem, we find the horizontal distance $x = \sqrt{d^2 - h^2} = 5$ miles. Substituting the given values into the equation, we find $\frac{dd}{dt} = 486$ miles per hour.