A rectangular sheet of metal of area of $1080$ m$^2$ is folded form a shelter as shown in figure

Question

A rectangular sheet of metal of area of $1080$ m$^2$ is folded form a shelter as shown in figure $A$. The dimension of the cross section is shown in figure $B$. $ABCD$ is a rectangle with $AB = x$ m and $BCD$ is an isosceles triangle with $BC = CD = 2x$ m. $\angle CBD = \angle CDB = \theta$ rad.

Given that the length of the shelter is $60$ m,

1. find the value of $x$.
2. show that $BD = 12 \cos \theta$.
3. show that the internal volume of the shelter, $V$ m$^3$, is given by $V = 1080(\sin 2\theta + 2 \cos \theta)$.
Hence, find in exact form, the value of $\theta$ for which $V$ is a maximum and the maximum value of $V$.

Accepted Answer

(a)(i) $x=3$ (ii) $BD=12\cos\theta$ (iii) $V=1080(\sin2\theta+2\cos\theta)$ (b) $\theta=\frac{\pi}{6},\quad V_{\max}=1620\sqrt3\text{ m}^3$

Differentiation - Differentiation of Trigonometric, Logarithmic and Exponential Functions - General

Differentiation - Differentiation of Trigonometric, Logarithmic and Exponential Functions - General

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