Sure, let's solve this step by step.
We know that the rate of decrease of the angle of the sun's elevation, d_θ/d_t, is -1 rad/hr. We also know that the height of the building, H is 400 feet.
The length of shadow, L, and the angle of elevation, θ, follows the basic trigonometry rule, tan(θ) = H / L. From this, we can derive that L = H / tan(θ).
We'll now differentiate both sides of this equation with respect to t:
dL/dt = -H * (sec^2(θ)) * d_θ/d_t
Now, at the point in time we're interested, let's assume θ = 45 degrees or π / 4 radians. In trigonometry, we know that sec^2(45) = 2. So,
dL/dt = -H * 2 * d_θ/d_t.
We substitute the given values into this equation. We have H=400 feet, and d_θ/d_t = -1 rad/hr. So,
dL/dt = -400 * 2 * -1 = 800 feet/hour.
Therefore, at the moment when the angle of elevation is 45 degrees, the shadow of a 400-feet building lengthens at a rate of 800 feet per hour as the angle of elevation of the sun decreases at a rate of 1 radian per hour.