A toy made in the shape of a hemisphere of diameter 7 cm surmounted by a cone. If this 15.5 cm high toy is polished at 20 paise per cm2, then find the cost of polishing.
(Take π = 22/7)
₹ 42.90
Understanding the Toy's Shape and Surface Area for Polishing
The toy is described as a hemisphere surmounted by a cone. This means a cone is placed on top of a hemisphere, with their bases joined together. When we polish the toy, we polish the outer surfaces that are exposed. These surfaces are the curved surface area of the hemisphere and the curved surface area of the cone.
Breaking Down the Toy's Dimensions
We are given the following information:
- The toy is in the shape of a hemisphere of diameter 7 cm surmounted by a cone.
- The total height of the toy is 15.5 cm.
- The polishing cost is 20 paise per cm<sup>2</sup>.
- We need to use \(\pi = 22/7\).
From the diameter of the hemisphere, we can find its radius:
Radius of hemisphere (r) = Diameter / 2 = 7 cm / 2 = 3.5 cm
Since the cone is surmounted on the hemisphere, the base radius of the cone is the same as the radius of the hemisphere.
Radius of cone base (r) = 3.5 cm
The total height of the toy is the sum of the height of the hemisphere and the height of the cone.
The height of the hemisphere is equal to its radius.
Height of hemisphere = r = 3.5 cm
Total height of toy = Height of hemisphere + Height of cone
15.5 cm = 3.5 cm + Height of cone
Height of cone (h) = 15.5 cm - 3.5 cm = 12 cm
Calculating the Surface Area to be Polished
The area to be polished is the sum of the curved surface area (CSA) of the hemisphere and the curved surface area of the cone.
Total Area = CSA of Hemisphere + CSA of Cone
Curved Surface Area of the Hemisphere
The formula for the curved surface area of a hemisphere is \(2\pi r^2\).
Using \(r = 3.5\) cm and \(\pi = 22/7\):
\(CSA_{hemisphere} = 2 \times \frac{22}{7} \times (3.5)^2\)
\(CSA_{hemisphere} = 2 \times \frac{22}{7} \times (3.5 \times 3.5)\)
\(CSA_{hemisphere} = 2 \times \frac{22}{7} \times 12.25\)
We can write 3.5 as 7/2:
\(CSA_{hemisphere} = 2 \times \frac{22}{7} \times (\frac{7}{2})^2 = 2 \times \frac{22}{7} \times \frac{49}{4}\)
\(CSA_{hemisphere} = 2 \times 22 \times \frac{7}{4}\) (cancelling 7 with 49)
\(CSA_{hemisphere} = 44 \times \frac{7}{4}\) (multiplying 2 and 22)
\(CSA_{hemisphere} = 11 \times 7\) (cancelling 4 with 44)
\(CSA_{hemisphere} = 77\) cm<sup>2</sup>
Curved Surface Area of the Cone
The formula for the curved surface area of a cone is \(\pi r l\), where 'l' is the slant height.
First, we need to find the slant height (l) using the radius (r) and height (h) of the cone. The relationship is given by the Pythagorean theorem: \(l^2 = r^2 + h^2\).
Using \(r = 3.5\) cm and \(h = 12\) cm:
\(l^2 = (3.5)^2 + (12)^2\)
\(l^2 = 12.25 + 144\)
\(l^2 = 156.25\)
To find 'l', we take the square root of 156.25:
\(l = \sqrt{156.25}\)
We can calculate this: \(12.5 \times 12.5 = 156.25\).
So, the slant height \(l = 12.5\) cm.
Now, we can calculate the CSA of the cone using \(r = 3.5\) cm, \(l = 12.5\) cm, and \(\pi = 22/7\):
\(CSA_{cone} = \pi r l\)
\(CSA_{cone} = \frac{22}{7} \times 3.5 \times 12.5\)
We can write 3.5 as 7/2:
\(CSA_{cone} = \frac{22}{7} \times \frac{7}{2} \times 12.5\)
\(CSA_{cone} = 11 \times 12.5\) (cancelling 7 and dividing 22 by 2)
\(CSA_{cone} = 137.5\) cm<sup>2</sup>
Total Area for Polishing
Total Area = CSA of Hemisphere + CSA of Cone
Total Area = \(77\) cm<sup>2</sup> + \(137.5\) cm<sup>2</sup>
Total Area = \(214.5\) cm<sup>2</sup>
Calculating the Total Polishing Cost
The polishing cost is 20 paise per cm<sup>2</sup>. First, convert this cost to rupees.
1 Rupee = 100 paise
20 paise = 20 / 100 Rupees = ₹ 0.20
Cost of polishing per cm<sup>2</sup> = ₹ 0.20
Total cost of polishing = Total Area \(\times\) Cost per cm<sup>2</sup>
Total cost = \(214.5 \times 0.20\)
Total cost = \(214.5 \times \frac{20}{100}\)
Total cost = \(214.5 \times \frac{1}{5}\)
Total cost = \(\frac{214.5}{5}\)
Total cost = \(42.9\)
The total cost of polishing is ₹ 42.90.
Summary of Calculations
| Item | Value | Calculation / Formula |
|---|---|---|
| Hemisphere Diameter | 7 cm | Given |
| Hemisphere Radius (r) | 3.5 cm | Diameter / 2 |
| Toy Total Height | 15.5 cm | Given |
| Hemisphere Height | 3.5 cm | Equals radius |
| Cone Height (h) | 12 cm | Total Height - Hemisphere Height |
| Cone Base Radius (r) | 3.5 cm | Same as hemisphere radius |
| Cone Slant Height (l) | 12.5 cm | \(\sqrt{r^2 + h^2}\) |
| CSA of Hemisphere | 77 cm<sup>2</sup> | \(2\pi r^2\) |
| CSA of Cone | 137.5 cm<sup>2</sup> | \(\pi r l\) |
| Total Area to Polish | 214.5 cm<sup>2</sup> | CSA Hemisphere + CSA Cone |
| Polishing Cost per cm<sup>2</sup> | ₹ 0.20 | 20 paise = ₹ 0.20 |
| Total Polishing Cost | ₹ 42.90 | Total Area \(\times\) Cost per cm<sup>2</sup> |
Revision Table: Key Concepts for Surface Area and Volume
| Shape | Curved Surface Area (CSA) | Total Surface Area (TSA) | Volume |
|---|---|---|---|
| Hemisphere (radius r) | \(2\pi r^2\) | \(3\pi r^2\) | \(\frac{2}{3}\pi r^3\) |
| Cone (radius r, height h, slant height l) | \(\pi r l\) | \(\pi r (r+l)\) | \(\frac{1}{3}\pi r^2 h\) |
In this problem, we only needed the curved surface areas because the bases of the hemisphere and cone were joined internally and were not polished.
Additional Information: Composite Solids and Surface Area
Problems involving composite solids (shapes made by combining two or more basic solids) often require you to identify which surfaces are exposed and need to be included in the surface area calculation. For polishing or painting, it's usually the outer visible surfaces.
Steps to solve problems on composite solids surface area:
- Identify the basic shapes forming the composite solid.
- Determine the dimensions of each basic shape (radius, height, slant height, etc.).
- Identify which parts of the basic shapes contribute to the total surface area of the composite solid (usually curved surfaces and any exposed base/top).
- Calculate the area of each contributing surface.
- Sum up the areas to find the total surface area.
- If a cost is involved, multiply the total area by the given rate per unit area.
Remember to pay close attention to units and convert them if necessary (like paise to rupees here).
Important Questions from Geometry
The angles of a cyclic quadrilateral, taken in order, are x°, (3x - 30)°, (y + 30)°, and (2x - y)°. Find the measure of the smallest angle of the quadrilateral.
If 2cosθ = √3, then what is the value of tan 2θ?
Length of three sides of a triangular field are 15m, 19m, and 22m respectively. What is the area of the field? (correct to one decimal place)
A triangle with vertices (3,1), (-1,0), (2,5) is:
If cos (x−y) = √3/2 and sin (x + y) = 1, where x > y, then the value of y is:
