Question

Find the area of the shade region if d=18 units. Round to the nearest tenth.

Answer 1

Area of shaded region = 254.5 square units

Explanation:

We have 3 semicircles:

d = 18 units

Semicircle 1 = The big semicircle

Semicircle 2 and 3 = The two small semicircles

Semicircle 2 = 18 units

Semicircle 3 = 18 units

Semicircle 1 = Semicircle 2 + Semicircle 3 = 36 units

Radius = Half of the diameter

Semicircle 1 radius = 18 units

Semicircle 2 radius = 9 units

Semicircle 3 radius = 9 units

`\mathrm{Area\:of\:a\:semicircle = ( \pi r^2 )/( 2 )}`

`\mathrm{Area\:of\:shaded\:region = Area\:of\:Semicircle\:1 - (Area\:of\:Semicircle\:2 + Area\:of\:Semicircle\:3)}`

`\mathrm{Area\:of\:shaded\:region = ( \pi r^2 )/( 2 ) - ( ( \pi r^2 )/( 2 ) + ( \pi r^2 )/( 2 ) )}`

`\mathrm{Substitute\:the\:numbers\:into\:the\:equation}`

`\mathrm{Area\:of\:shaded\:region = ( \pi (18)^2 )/( 2 ) - ( ( \pi (9)^2 )/( 2 ) + ( \pi (9)^2 )/( 2 ) )}`

`\mathrm{Do\:all\:of\:the\:exponents\:first}`

`\mathrm{Area\:of\:shaded\:region = ( \pi*324 )/( 2 ) - ( ( \pi*81)/( 2 ) + ( \pi*81)/( 2 ) )}`

`\mathrm{Combine\:( \pi*81)/( 2 )\:and\:( \pi*81)/( 2 )\:to\:get\:\pi*81}`

`\mathrm{Area\:of\:shaded\:region = ( \pi*324 )/( 2 ) - ( \pi*81 )}`

`\mathrm{Divide\: \pi *324\:by\:2}`

`\mathrm{Area\:of\:shaded\:region = { \pi*162} - \pi*81}`

`\mathrm{Combine\:\pi *162\:and\:- \pi *81}`

`\mathrm{Area\:of\:shaded\:region = 81\pi}`

`\mathrm{Area\:of\:shaded\:region = 254.469004941}`

Area of shaded region rounded to the nearest tenth is: 254.5 square units

look at the picture for more information

Answer 2

29.3 units^2

Explanation:

sorry for no explanation