Garden Shed
Can you minimise the amount of wood needed to build the roof of my garden shed?
Age
11 to 14
Challenge level
Problem
Suppose we have a garden shed with a roof made from five wooden beams laid out like this:
Image
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 $4x167.3+100=769.2cm$ 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 legnth 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, and used calculus to find the minimum value this function takes. Here is a link to Niharika's solution.
Image
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 $4x167.3+100=769.2cm$ 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 legnth of wood need is $764.758$ and the value of $x$ needed for this is approximately $77.5cm$.
Image
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, and used calculus to find the minimum value this function takes. Here is a link to Niharika's solution.