a satellite camera takes a rectangular - shaped picture. the smallest region that can be photographed is a 4 - km by 6 - km rectangle. as the camera zooms out, the length l and width w of the rectangle increase at a rate of 3km/sec. how long does it take for the area A to be at least 4 times its original size?
2.424
To solve this problem, we need to determine how long it takes for the area (A) of the rectangle to be at least 4 times its original size.
First, let's find the original area (A₀) of the rectangle. The original length (l₀) of the rectangle is 4 km and the original width (w₀) is 6 km.
A₀ = l₀ * w₀
A₀ = 4 km * 6 km
A₀ = 24 km²
Next, we need to find the rate of change of area with respect to time. We know that both the length (l) and the width (w) of the rectangle are increasing at a rate of 3 km/sec.
So, the rate of change of area (dA/dt) is given by:
dA/dt = dw/dt * l + dw/dt * w
We substitute the given values:
dA/dt = 3 km/sec * 4 km + 3 km/sec * 6 km
dA/dt = 12 km²/sec + 18 km²/sec
dA/dt = 30 km²/sec
The rate of change of area is 30 km²/sec.
Now, let's denote the new area as Aₙ and the time it takes for the area to increase by 4 times as t.
Aₙ = A₀ + (dA/dt) * t
Since we want the area to be at least 4 times its original size, we can set up the following equation:
Aₙ = 4A₀
A₀ + (dA/dt) * t = 4A₀
Substituting the values we previously found:
24 km² + 30 km²/sec * t = 4 * 24 km²
54 km²/sec * t = 3 * 24 km²
54 km²/sec * t = 72 km²
Dividing both sides of the equation by 54 km²/sec:
t = 72 km² / 54 km²/sec
t ≈ 1.33 sec
Therefore, it will take approximately 1.33 seconds for the area to be at least 4 times its original size.