A road running north to south crosses a road going east to west at the point P. Car A is driving north along the first road, and an airplane is flying east above the second road. At a particular time the car is 15 kilometers to the north of P and traveling at 55 km/hr, while the airplane is flying at speed 175 km/hr 10 kilometers east of P at an altitude of 2 km. How fast is the distance between the car and the airplane changing?
Would you use the derivative of the pathagorium theorem? How would you go about doing this?
D^2 = Pn^2 + Pe^2 + 2^2
use implicit differentiation ... 2 D dD/dt = 2 Pn dPn/dt + 2 Pe dPe/dt
√(15^2 + 10^2 + 2^2) dD/dt = 15 * 55 + 10 * 175
But when I did this, I got 141.96 which according to the database I'm entering this into, Is still incorrect.
141.96 may have too many significant figures
try rounding to three digits
To find out how fast the distance between the car and the airplane is changing, we can indeed use the derivative of the Pythagorean theorem.
Let's assign some variables:
- Let x be the distance between the car and point P (along the north-south road).
- Let y be the distance between the airplane and point P (along the east-west road).
- Let d be the distance between the car and the airplane.
We want to find the rate at which d changes with respect to time (t), which is denoted as dx/dt.
Using the Pythagorean theorem, we can write an equation relating x, y, and d:
d^2 = x^2 + y^2
Differentiating both sides of the equation with respect to time (t), we get:
(2d)(dd/dt) = (2x)(dx/dt) + (2y)(dy/dt)
Simplifying the equation, we have:
dd/dt = (x*dx/dt + y*dy/dt) / d
Now, let's substitute the given values:
dx/dt = 55 km/hr (rate of change of x, the car's distance to P)
dy/dt = 0 km/hr (the airplane is flying at a constant eastward speed, so its distance to P is not changing)
x = 15 km (the car is 15 km to the north of P)
y = 10 km (the airplane is 10 km east of P)
Plugging in these values, we have:
dd/dt = (15 * 55 + 10 * 0) / d
We need to find the value of d to calculate dd/dt. Using the Pythagorean theorem again:
d^2 = x^2 + y^2
d^2 = 15^2 + 10^2
d^2 = 225 + 100
d^2 = 325
d = sqrt(325) ≈ 18.03 km
Now, substituting the value of d into our equation:
dd/dt = (15 * 55) / 18.03
dd/dt ≈ 45.2 km/hr
Therefore, the distance between the car and the airplane is changing at a rate of approximately 45.2 km/hr.