A bucket made up of a metal sheet is in the form of 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 bucket if the cost of metal sheet used is Rs 15 per 100 cm2. [Use π = 3.14.]
Given: height of container frustum = h = 16 cm
Radius of lower circular end = r = 8 cm
Radius of upper circular end = R = 20 cm
Cost of 100 cm2 metal sheet = 15 Rs
∴ Cost of 1 cm2 metal sheet = 15/100 = 0.15 Rs
Formula: Total surface area of frustum = πr2 + πR2 + π(R + r)l cm2
Where l = slant height
∴ l = 20 cm
Since it is given that a bucket is to be made hence the top is open we need to subtract the area of top/upper circle from total surface area of frustum because we don’t require a metal plate for top.
Radius of top/upper circle = R
Area of upper circle = πR2
∴ area of metal sheet used = (total surface area of frustum)-πR2
= πr2 + πR2 + π(R + r)l- πR2 cm2
= πr2 + π(R + r)l cm2
= π × (82 + (20 + 8)20) cm2
= 3.14 × 624 cm2
= 1959.36 cm2
∴ 1959.36 cm2 metal sheet is required to make the container.
∴ Cost of 1959.36 cm2 metal sheet = 1959.36 × cost of 1 cm2 metal sheet
= 1959.36 × 0.15 Rs
= 293.904 Rs
∴ Cost of metal sheet required to make container = 293.904 Rs