Question

A container, opened from the top and made up of a metal sheet, is in the form of a frustum of a cone of height 16 cm with radii of its lower and upper ends as 8 cm and 20 cm, respectively. Find the cost of the milk which can completely fill the container, at the rate of Rs 50 per litre. Also find the cost of metal sheet used to make the container, if it costs Rs 10 per 100 cm2.

Answer 1

Answer:

Step-by-step explanation:

Given the height of the container $(h)= 16\text{ cm}</p><p>$

Radius of the upper end $R_{1}=20\text{ cm}$

Radius of the lower end $R_{2}=8\text{ cm}$

The shape of the bucket is frustum

The slant height of the frustum

$S=\sqrt{h^{2}+(R_{1}-R{2})^{2}}\\\\\Rightarrow{S}=\sqrt{16^{2}+(20-8)^{2}}\\\\\Rightarrow{S}=\sqrt{16^{2}+12^{2}}\\\\\Rightarrow{S}=\sqrt{256+64}\\\\\Rightarrow{S}=\sqrt{320}\\\\\Rightarrow{S}=17.88\text{ cm}$

The slant height of the frustum $S=17.88\text{ cm}$

The surface are of the container = Lateral Area of a flat surface + Circular area of bottom

therefore,

$A=\pi\times(R_{1}+R_{2})\times{S}+\pi\times({R}_{1})^{2}\\\\\Rightarrow{A}=\frac{22}{7}\times(20+8)\times17.88+\frac{22}{7}\times(20)^{2}\\\\\Rightarrow{A}=\frac{22}{7}\times28\times17.88+\frac{22}{7}\times(20)^{2}\\\\\Rightarrow{A}=\frac{22}{7}(500.64+400)\\\\\Rightarrow{A}=\frac{22}{7}\times(900.64)\\\\\Rightarrow{A}=2830.5$

therefore the total surface area of a bucket $A=2830.5\text{ cm}^{2}}$

Cost of metal sheet for $100\text{ cm}^2$ is 10 rupees.

therefore, the Cost of aluminium sheet for $2830.5\text{ cm}^{2}}$ is $=\frac{10}{100}\times2830.5=283.05$  

The cost of the aluminium sheet is rupees 283.05

Volume of the container $V=\frac{1}{3}\times\pi\timesh\times(R_{1}^{2}+R_{1}R_{2}+R_{2}^{2})$

Therefore,

$V=\frac{1}{3}\times\pi\timesh\times(R_{1}^{2}+R_{1}R_{2}+R_{2}^{2})\\\\\Rightarrow{V}=\frac{1}{3} \times\frac{22}{7}\times16\times(20^{2}+20\times8+8^{2})\\\\\Rightarrow{V}=\frac{1}{3} \times\frac{22}{7}\times16\times(400+160+64)\\\\\Rightarrow{V}=\frac{1}{3} \times\frac{22}{7}\times16\times624\\\\\Rightarrow{V}=10459.42$

Volume of the container $V==10459.42\text{cm}^{2}=10.45942\text{ liter}$

cost of the milk Rs 50 per liter

Total cost of the milk $=50\times10.45942=522.97=523\text{ approx}$