An airplane, flying horizontally at an altitude of 1 mile, passes directly over an observer. If the constant speed of the airplane is 400 miles per hour, how fast is its distance from the observer increasing 45 seconds later? Hint: Note that in 45 seconds, the airplane goes 5 miles.
A metal disk expands during heating. If its radius increases at the rate of 0.02 inch per second, how fast is the area of one of its faces increasing when its radius is 8.1 inches?
totally don't know the idea in these questions, help, please!
Answer
This smells like homework ( ;-) ), so I'm just going to get you set up. If you need more help, leave a comment.
The first problem is best accompanied by a picture. The second doesn't really need one, as the picture is trivial. I'll address the second problem first, as it is somewhat easier.
The Second Problem
The rate of change of the radius is $0.02\frac{in}{s}$. Thus:
$$\frac{dr}{dt} = 0.02$$
Area of circle is easy:
$$A = \pi r^2$$
Differentiate area:
$$\frac{dA}{dt} = \pi((2r)\frac{dr}{dt})$$
All that remains is to plug in the radius and rate of change of radius.
The First Problem
(If the picture doesn't show up, let me know... imgur may have been blocked on my computer.)
$x$ is the horizontal distance the plane has traveled since directly passing over the observer. $d$ is the distance you are seeking. (oops--just realized $d$ won't work as a symbol very well for obvious reasons. I'm going to substitute $r$.)
$$\frac{dx}{dt} = velocity = 400$$
What is the distance?
$$r = \sqrt{x^2 + 1^2}$$
$$\frac{dr}{dt} = \frac{(2x)\frac{dx}{dt}}{2\sqrt{x^2 + 1}}$$
Now you just have to plug in the values for $x$ and $\frac{dx}{dt}$.
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