Question

The shaded part of the diagram below shows the top surface of an engine part.

5
Diagram not drawn to scale
The measurements taken by a motor engineer are:
• reflex angle BÔC = 240°,
• AO=OD= 6 cm,
• BO=OC=3 cm.
The top surface of the engine part is to be painted.
The paint costs 15p per cm².
75
Using = 3, calculate how much it costs to paint the top surface of 20 engine parts.
Give your answer in pounds.
(1 mark)

Answer 1

The cost to paint the top surface of 20 engine parts is £135.

To calculate the cost of painting the top surface of 20 engine parts, we first need to find the total area of the top surface of one engine part, and then multiply it by 20 to account for the 20 engine parts. The total cost can be calculated using the following steps:

Find the total area of the top surface of one engine part.

Multiply the total area by 20 to account for the 20 engine parts.

Multiply the total area by the cost per cm², which is 15p, and then convert the result to pounds.

Let's proceed with the calculation:

The total area of the top surface of one engine part can be found using the measurements provided. The top surface is composed of two squares with side lengths of 6 cm and 3 cm. The total area (A) is the sum of the areas of the two squares:

A=(6×6)+(3×3)

A=36+9

A = 45 cm².

Multiply the total area by 20 to account for the 20 engine parts:

Total Area=45cm² x 20

Total Area=900cm²

Multiply the total area by the cost per cm², which is 15p, and then convert the result to pounds. To convert from pence to pounds, divide by 100:

Cost = 900 cm² × 15р/ст² 2

Cost = 13500p

Cost = £135 (since100p = £1)

Therefore, the cost to paint the top surface of 20 engine parts is £135.

Answer 2

To calculate the cost of painting the top surface of 20 engine parts, we need to find the surface area of one engine part and then multiply it by 20.

To find the surface area, we need to find the length of the curved surface. We can do this by using the Pythagorean theorem to find AB and then use that to find the circumference of the curved surface.

AB^2 = BO^2 + AO^2 = 9 + 36 = 45

AB = sqrt(45) = 3 sqrt(5)

Circumference = 2 * pi * AB = 2 * pi * 3 sqrt(5) = 6 pi sqrt(5)

Using the fact that the reflex angle BÔC is 240 degrees, we can find the length of the curved surface:

240° / 360° = 2/3 of the circumference

Curved surface length = 2/3 * 6 pi sqrt(5) = 4 pi sqrt(5)

Finally, we can find the surface area of one engine part:

Surface area = (2 * 6 * 3) + (4 pi sqrt(5) * 6) = 36 + (24 pi sqrt(5))

To paint 20 engine parts, the cost would be:

20 * surface area * 15p = 20 * 36 + 20 * 24 pi sqrt(5) * 15p = 720 + (24 * 20 * 15 * pi * sqrt(5)) p = 720 + 720 pi sqrt(5) p

Since 1 pound is equal to 100 pence, the cost in pounds would be:

720 + 720 * pi * sqrt(5) / 100 = 720 + 720 * 0.177 / 100 = 720 + 1.264

So, the cost to paint the top surface of 20 engine parts would be 721.264 pounds.