Looking please for any help! The density of marble used for the pedestal shown in the figure below is $2750\text{ kg/m}^3$
. A crane can lift a maximum of 2 tons. Can the crane raise this pedestal?
My attempt was the following: To find the volume of the frustum: I use the formula of the trapezoid surface: $$\frac{(1+0.4)\cdot 1.5}{2}=1.05\text{ m}^2$$
And then $1.05\times 1.5=1.575\text{ m}^3\longrightarrow$ the mass is $1.575\times 2750=4331\text{ kg}=4.3\text{ tons}$
And my answer was the crane can’t raise the pedestal.
I don’t know if this is the right answer! All help/solutions appreciated. Thanks
Best Answer
I will, as an A Level student, try to show you exactly what you'd need to understand to get full marks in a similar GCSE question. Firstly, before I begin, you need to realise that you have not actually found the volume of the solid.
You have found the cross sectional area of this solid when looked at perpendicular to its base. Indeed, this area is equal to $1.05\text{m}$. But this is not relevant to the question. We are looking for volume, not area or surface area.
Now to begin:
Throughout this solution I will assume that the bases of this truncated pyramid are square, which would be a more normal GCSE question.
Think of the solid as a larger pyramid with its top, a smaller pyramid, truncated. I will avoid trigonometry in case you have not covered it yet.
Let the height of the larger pyramid (which currently has its top chopped off) be $h$. Then the height of the smaller pyramid (which is chopped off) is $h-1.5$. Using similar triangles, we see that
$$\frac{h}{0.5}=\frac{h-1.5}{0.2}\implies0.2h=0.5h-0.75\implies h=\frac{0.75}{0.3}=2.5$$ So the height of the smaller pyramid is $2.5-1.5=1$. Now, you should know that the volume of a square based pyramid with base length $a$ and perpendicular height $h$ is $$\frac{1}{3}a^2h$$ Hence, the volume, $V_1$, of the smaller pyramid is $\frac{1}{3}(0.4^2)(1)=\frac{4}{75}$. Similarly, the volume of the larger pyramid, $V_2$, is $\frac{1}{3}(1^2)(2.5)=\frac{5}{6}$. Hence, the volume of the solid we are dealing with is $$ V_2-V_1=\frac{5}{6}-\frac{4}{75}=0.78\text{m}^3$$ Using $$\text{density}=\frac{\text{mass}}{\text{volume}}$$ we can rearrange to find the required mass: $$\text{mass}=\text{density}\times\text{volume}=0.78\times2750=2145\text{kg}$$ As $2145\text{kg}$ is larger than $2$ tonnes, we conclude that the crane cannot lift the pedestal.
I hope I have helped you. If you have any further queries, please don't hesitate to ask. I wish you luck in whatever assessments your school give you in place of GCSE's.
Please feel free to email me at
[email protected]if you have any other problems. I will be glad to help you.