The perimeter of the rectangle as a function of w only is P(w) = 2[√(324 - w²) + w]
The perimeter of the rectangle P = 2(h + w)
Since the diagonal is D = 18 feet, using Pythagoras' theorem, we have that
D² = h² + w²
So, making h subject of the formula, we have
h = √(D² - w²)
h = √(18² - w²)
h = √(324 - w²)
Since the perimeter of the rectangle P = 2(h + w), substituting the value of h into P, we have
P = 2(h + w)
P = 2[√(324 - w²) + w]
P(w) = 2[√(324 - w²) + w]
So, the perimeter of the rectangle as a function of w only is P(w) = 2[√(324 - w²) + w]
The angle θ as a function of w only is θ(w) = sin⁻¹[w/18]
Using trigonometric functions, sinθ = w/D
since D = 18, substituting the value of D into the equation, we have
sinθ = w/18
Taking inverse tan of both sides, we have
θ = sin⁻¹[w/18]
θ(w) = sin⁻¹[w/18]
So, the angle θ as a function of w only is θ(w) = sin⁻¹[w/18]
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