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Answer 1

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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