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McGraw Hill Integrated II, 2012 View details

5. Volumes of Pyramids and Cones

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Exercise 20 Page 845

Use the formula for the volume of a cone.

3.1ft^3

Practice makes perfect

To calculate the volume of a cone, we can use the following formula. In this formula r is the radius and h is the height of a cone. V= 13π r^2 h We are given that the slant height of the cone is 2ft and an angle between the slant height and radius is 47^(∘). Let's notice that the slant height, radius, and the height form a right triangle.

The trigonometric function which connects height and the slant height is the sine. sin θ =opposite/hypotenuse Let's substitute the values and calculate the height.

sinθ=opposite/hypotenuse

SubstituteValues

Substitute values

sin 47^(∘)=h/2

▼

Solve for h

MultEqn

LHS * 2=RHS* 2

2sin47^(∘)=h

RearrangeEqn

Rearrange equation

h=2sin47^(∘)

UseCalc

Use a calculator

h=1.4627...

RoundDec

Round to 1 decimal place(s)

h≈ 1.5

The height of the cone is approximately 1.5 ft. Now let's use a fact that the trigonometric function which connects the radius and the slant height is the cosine. cosθ =adjacent/hypotenuse Let's substitute the values and calculate the radius.

cos(θ )=adjacent/hypotenuse

SubstituteValues

Substitute values

cos47^(∘)=r/2

▼

Solve for r

MultEqn

LHS * 2=RHS* 2

2cos47^(∘)=r

RearrangeEqn

Rearrange equation

r=2cos47^(∘)