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McGraw Hill Integrated II, 2012

MH

McGraw Hill Integrated II, 2012 View details

6. Surface Areas and Volumes of Spheres

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Exercise 16 Page 852

A hemisphere is half of a sphere.

To find the area of a hemisphere we need to calculate the area of half of a sphere with the same radius and add the area of the great circle. Surface Area of a Hemisphere [0.8em] A=1/2(4π r^2)+π r^2 We first need to calculate the radius. To do so, we will consider the given fact that the area of a circle A of the great circle is 40 in^2. Let's recall the formula for the area. A=π r^2 We can substitute 40 for A in this formula and solve for the radius r.

A=π r^2

A= 40

40=π r^2

▼

Solve for r

.LHS /π.=.RHS /π.

40/π=r^2

Rearrange equation

r^2=40/π

Use a calculator

r^2=12.73...

Round to 1 decimal place(s)

r^2≈ 12.7

sqrt(LHS)=sqrt(RHS)

r≈ 3.56

Round to 3.6 decimal place(s)

r≈ 3.6

We know that r≈ 3.6in. We can substitute this value in the formula for the area of a hemisphere and simplify.

A=1/2(4π r^2)+π r^2

r= 3.6

A=1/2(4π ( 3.6^2))+π ( 3.6^2)

▼

Evaluate right-hand side

Calculate power

A=1/2(4π (12.96))+π (12.96)

CommutativePropMult

Commutative Property of Multiplication

A=1/2(4(12.96)π )+12.96π

Multiply

A=1/2(51.84π )+12.96π

MoveRightFacToNumOne

1/b* a = a/b

A=51.84/2π+12.96π

Calculate quotient

A=25.92π+12.96π

Add terms

A=38.88π

Use a calculator

A=122.145...

Round to 1 decimal place(s)

A≈ 122.2

The area of the hemisphere to the nearest tenth is 122.2in^2.

Subchapter links

Exercises p.852-855