Question 1 · Making tin cans
A cylinder with given volume and radius: the height, the tin needed, the cost and the budget.
Maximum score · 5 pointsSara wants to start a tin can manufacturing company. Each can must have the shape in the figure: a volume of $141.4\ \text{cm}^3$ and a radius of $3$ cm.
To make the cans she must use sheets of tin. She has contacted a tin factory that sells it at €$4.45$ per $\text{m}^2$. Sara has to make $5000$ cans every month and has a monthly budget of €$400$ to buy the tin sheets she needs.
- How many square metres does Sara need to produce each can? 2 p
- How many € will it cost to make each can? (If you could not find the previous area, you may use this data: each can has an area of $175\ \text{cm}^2$.) 1.5 p
- Will she be able to buy all the tin she needs to make all the cans with her budget, or will she need a loan? (If you could not find the previous area, you may use this data: making each can costs €$0.05$.) 1.5 p
Step-by-step solution
a)
First we find the height from the cylinder volume: $V = \pi r^2 h$.
The tin needed is the total area of the cylinder, $A_T = 2A_B + A_L$:
Since the price is per $\text{m}^2$, we convert to square metres ($1\ \text{m}^2 = 10\,000\ \text{cm}^2$):
b)
We multiply the area in $\text{m}^2$ by the price of tin:
With the alternative data: $0.0175 \cdot 4.45 \approx 0.078$ €.
c)
Monthly cost of the $5000$ cans:
With the alternative data: $5000 \cdot 0.05 = 250$ €, not needed either.