Figure I shows a shelter made by a folded rectangular metal sheet of a fixed area of $1080$ m\(^2\

Question

Figure I shows a shelter made by a folded rectangular metal sheet of a fixed area of $1080$ m$^2$, where the length of the shelter is $60$ m. Figure II shows the cross section of the shelter, where $ABDE$ is a rectangle with $AB = DE = x$ m and triangle $BCD$ is isosceles with $BC = CD = 2x$ m. Let angle $CBD =$ angle $CDB = \theta$ radians. State the range of values of $\theta$. Find the value of $x$. Show that the internal volume of the shelter, $V$ m$^3$, is given by $V = 1080(\sin 2\theta + 2\cos \theta)$. Find the exact value of $\theta$ for which $V$ has a stationary value. Hence find the stationary value of $V$ and determine its nature. Give the exact value.

Accepted Answer

(i) $0<\theta<\dfrac{\pi}{2}$ (ii) $x=3\text{ m}$ (iii) $V=1080(\sin2\theta+2\cos\theta)\text{ m}^3$ (iv) $\theta=\dfrac{\pi}{6}$ (v) $1620\sqrt3\text{ m}^3$, maximum

Techniques of Differentiation - Derivatives of Trigonometric Functions

Techniques of Differentiation - Derivatives of Trigonometric Functions

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