Garden shed
Can you minimise the amount of wood needed to build the roof of my garden shed?
Problem
Each of the four sloping beams is the same length.
The roof of the shed is 300cm long, 240cm wide and 60cm high.
What is the total length of wood needed to make these five main beams for the roof?
Can you reduce the amount of wood needed by changing the lengths marked in red?
With thanks to Don Steward, whose ideas formed the basis of this problem.
Student Solutions
Together, the beams of wood used to make the shed roof measure $780cm$. Here's a really clear solution diagram that Hamza from Hitchin Boys' School made:
The second part of the question asked whether we could use less wood by changing the red length of $120cm$. Lots of people who thought about this question used different methods.
Sean from Sacred Heart Catholic College tried some different values for this red length to see if they used less wood:
If the red length was $100cm$ the middle beam would be $100cm$ also, we can tell this using the bird's eye view.
Then, using the same technique as the first part of the question with triangles and Pythagoras' theorem I discovered the sloping beams are $167.3cm$.
So a roof with red length $100cm$ would require $4\times 167.3+100=769.2cm$ of
wood; $10.8cm$ less than that needed for a roof with red length $120cm$.
Eleanor thought about the length of wood needed as a function of the red length, and drew a graph of this function:
Let the length of the red line be $x$. To find the shortest possible length, the best way is to plot a graph of the points for the equation $y=\sqrt{x ²+18000} \times 4 + 300-2x$. This is the total length of wood needed when the red length is $x$. I used an app to plot my graph and give me the table, screenshots below. The table of results shows that the shortest possible length of wood need is $764.758$ and the value of $x$ needed for this is approximately $77.5cm$.
Joshua from QEGS Penrith and Niharika from Rugby School also thought about the total length of wood needed as a function of the red length, but used calculus, which is more advanced mathematics, to find the minimum value this function takes. Part of Joshua's solution is below:
The formula for the length of wood required is $300 - 2x + 4\sqrt(x^2 + 18000)$ (where $x$ is the red length).
To find the minimum value of $x$ and therefore the total length of wood we must first differentiate the equation of the total length of wood.
The differentiated equation is $$-2 + \dfrac{4x}{\sqrt{x^2 + 18000}}$$, this comes from the fact that the differential of $300$ is $0$, the differential of $-2x$ is $-2$ and, using the chain rule, we get that the differential of $4\sqrt{x^2 + 18000}$ is $4\times 2x\times \dfrac{1}{2}\sqrt{x^2 + 18000}^{-1}$ or just $\dfrac{4x}{\sqrt{x^2 + 18000}}$.
Setting this to zero and solving the equation for $x$, to obtain a minimum value of $x$, we get $x = 20\sqrt{15}$. Putting this into the equation we get 764.758 to 3dp.