Exercise 17 Page 828

Let's first calculate the lateral area and then use the result to calculate the surface area.

Lateral Area

The given solid is a cone.

To calculate the lateral area of a cone, we can use the known formula where r is the radius of the base and l is the slant height of the cone. L=π rl To find the radius we can use the Pythagorean Theorem. When doing this, the slant height l is the hypotenuse. The altitude a and the radius r of the cone are the legs. Let's use these given values to solve for r.

r^2+a^2=l^2

SubstituteII

a= 5, l= 9 12

r^2+ 5^2 = ( 9 12)^2

▼

Solve for r

MixedToFrac

a bc=a* c+b/c

r^2+5^2 = (19/2)^2

PowQuot

(a/b)^m=a^m/b^m

r^2+5^2 = 19^2/2^2

CalcPow

Calculate power

r^2 + 25 = 361/4

SubEqn

LHS-25=RHS-25

r^2 = 361/4-25

NumberToFrac

a = 4* a/4

r^2 = 361/4-100/4

SubFrac

Subtract fractions

r^2 = 261/4

SqrtEqn

sqrt(LHS)=sqrt(RHS)

r = sqrt(261/4)

UseCalc

Use a calculator

r = 8.077747...

RoundDec

Round to 4 decimal place(s)

r ≈ 8.0777

Note that we only kept the principal root when solving the equation because l is the radius of the base of a cone and it must be non-negative. We are also given that the slant height of the cone is 9 12ft. By substituting r= 8.0777 and l with 9 12 into the formula, we can calculate L.

L=π rl

SubstituteII

r= 8.0777, l= 9 12

L=π( 8.0777)( 9 12)

▼

Simplify right-hand side

WriteDec

Write as a decimal

L = π(8.0777)(9.5)

UseCalc

Use a calculator

L=241.080008...

RoundDec

Round to 1 decimal place(s)

L≈ 241.1

The lateral area of the cone is about 241.1ft^2.

Surface Area

To calculate the surface area of a pyramid, we need to calculate the sum of the lateral area L and the area of the base B. S=L+B Recall that the base of the cone is a circle. We can find its area using the formula B = π r^2.

B = π r^2

Substitute

r= 8.0777

B = π*( 8.0777)^2

▼

Simplify right-hand side

UseCalc

Use a calculator

B = 204.986524...

RoundDec

Round to 1 decimal place(s)

B≈ 205.0