Question - Calculating Total Length of Timber for Symmetrical Roof Truss
Solution:
The image provides a diagram of a symmetrical triangular roof truss with a pitch of 35°. It's given that $$ AC = 6 $$ meters, and we need to calculate the total length of timber required to make the truss, considering all the lengths $$ AD, FB, $$ and $$ BG $$ are equal, as well as $$ GC $$.Given the symmetry of the problem and the 35° pitch, we can break down the truss into different right-angled triangles and use trigonometric ratios to find the lengths of the sides.For the main triangle $$ ABC $$:- $$ AC $$ is the base and is given to be $$ 6 $$ meters.- $$ AB $$ can be found using the given pitch angle.The pitch of $$ 35° $$ tells us that angle $$ BAC $$ is $$ 35° $$. Since $$ ABC $$ is an isosceles triangle (because $$ AB = BC $$), angle $$ ABC = 35° $$ as well.In right-angled triangle $$ ABD $$:- $$ AD $$ (which is half of $$ AB $$) is the opposite side to angle $$ BAC $$ of $$ 35° $$.- $$ AC/2 $$ or $$ 3 $$ meters is the adjacent side to angle $$ BAC $$ of $$ 35° $$.Using the tangent function:$$\tan(35°) = \frac{AD}{3}$$Solving for $$ AD $$:$$AD = 3 \cdot \tan(35°)$$Since your truss is symmetrical and comprised of similar triangles, you can use this approach to calculate the total length of timber:- Length of $$ AD $$ (same as $$ FB $$, $$ BG $$, $$ GC $$) = $$ 3 \cdot \tan(35°) $$ meters- Total length for all of these 4 members is $$ 4 \cdot AD $$.In the smaller triangles $$ AFD $$, $$ BFE $$, and $$ BGC $$, the angle at $$ F $$ and $$ G $$ will still be $$ 35° $$. Given that each smaller triangle is half the size of the larger one, we can apply the same approach to find the length of $$ DF $$ (which will also equal $$ FE $$ and $$ GB $$):- $$ DF $$ (same as $$ FE $$, $$ GB $$; is half the size of $$ AD $$, so its length can be calculated as $$ \frac{AD}{2} $$) = $$ \frac{3 \cdot \tan(35°)}{2} $$ meters.- Total length for all of these 3 members is $$ 3 \cdot DF $$.Now calculate the actual values for $$ AD $$ and $$ DF $$ using a calculator for tangent of $$ 35° $$, and then add them together to find the total length of timber needed:$$\text{Total timber length} = 4 \cdot AD + 3 \cdot DF$$ Let's calculate the values using a calculator:$$AD = 3 \cdot \tan(35°) \approx 3 \cdot 0.7002 \approx 2.1006 \text{ meters}$$$$DF = \frac{3 \cdot \tan(35°)}{2} \approx \frac{3 \cdot 0.7002}{2} \approx 1.0503 \text{ meters}$$Now, let's find the total length of timber:$$\text{Total timber length} = 4 \cdot (2.1006) + 3 \cdot (1.0503) \approx 4 \cdot 2.1006 + 3 \cdot 1.0503 \approx 8.4024 + 3.1509 = 11.5533 \text{ meters}$$So, the total length of timber required to make the truss is approximately $$ 11.5533 $$ meters, depending on how accurate the approximation or calculator is.
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