A satellite camera takes a rectangular-shaped picture. The smallest region that can be photographed is a 4-km by 4-km rectangle. As the camera zooms out, the length l and width w of the rectangle increase at a rate of 3 km/sec. How long does it take for the area A to be at least 4 times its original size?

Question

20-05-2016
Mathematics

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A satellite camera takes a rectangular-shaped picture. The smallest region that can be photographed is a 4-km by 4-km rectangle. As the camera zooms out, the length l and width w of the rectangle increase at a rate of 3 km/sec. How long does it take for the area A to be at least 4 times its original size?

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Hagrid Hagrid · Answer 1 · 2016-06-02T01:42:04-04:00

So we know that the formula for the area of a rectangle is \(A = lw\).
Now both the length and width of the rectangle increase at 3 km/s, therefore, \(A(t) = (3t+l)*(3t+w). Since the initial length = initial width = 4 km, then the initial area = 16 [tex]km^2\). We want to know the time when the area is four times its original area, therefore, our new formula is: \(4A(t) = (3t+l)*(3t+w)\). Plugging in our known values we have:

\(64 [km^2] = (3t + 4 [km])*(3t + 4 [km])\)
\(t = \frac{4}{3} s\)

The area is four times its original area after \frac{4}{3} s[/tex].