Exercise 29 Page 72

Let's first calculate the surface area and then the volume of the fence post.

Surface Area

The surface of the post that Tara is going to paint consists of the surface of the pyramid's faces and the surface of the prism's faces. Let's find the area of each of these surfaces and then add them.

Surface Area of the Pyramid's Faces

We can calculate the surface area of a pyramid's faces using the formula below, where P is the perimeter of the base and l is the slant height. Note that we will not be using the full formula for surface area of a pyramid because Tara will not be painting the base of this figure. SA_(pyramid) (including the Base)=& 1/2Pl +B SA_(pyramid)(excluding the Base)=& 1/2Pl We are told that the prism is square, which means that its base is a square. From the diagram, we know that its side is 6 inches. Multiplying the side length by 4, we get the perimeter of the square. P=4* 6=24in Now, let's calculate the slant height. In order to do that, we need to draw a right triangle, one leg of which is the pyramid's height, inside the pyramid. We are given that the height of the pyramid is 4 inches. Take a look at the diagram below.

As we can see, the segment AB has the same length as the square's sides, 6 inches. The point C is the midpoint of the segment, so the segment BC has the length of BC=6/2=3inches. Since DC is the height of the pyramid, the triangle △ BCD is a right triangle. Therefore, we can use Pythagorean Theorem to calculate the slant height l. DC^2+BC^2=BD^2 We know that the length of the height DC equals 4 inches. Earlier we found that BC is equal to 3 inches. Also, BD is the length of the slant height l. Substituting these values into the formula, we get the equation for the unknown l. 4^2+ 3^2= l^2 Let's solve it!

4^2+3^2=l^2

▼

Solve for l

CalcPow

Calculate power

16+9=l^2

AddTerms

Add terms

25=l^2

SqrtEqn

sqrt(LHS)=sqrt(RHS)

5=l

RearrangeEqn

Rearrange equation

l=5

Thus, the slant height of the pyramid is 5 inches. Now, we know everything we need to find the surface area of the faces! Substituting P with 24 and l with 5 we get the following.

SA_(pyramid)=1/2Pl

SubstituteII

P= 24, l= 5

SA_(pyramid)=1/2* 24* 5

▼

Multiply

Multiply

Multiply

SA_(pyramid)=1/2* 120

MoveRightFacToNumOne

1/b* a = a/b

SA_(pyramid)=120/2

CalcQuot

Calculate quotient

SA_(pyramid)=60

Therefore, the surface area of the faces of the pyramid is 60 in^2.

Surface Area of the Prism's Faces

The surface area of the prism can be calculated using the formula below, where P is the perimeter of the base and h is the height of the prism. Once again we will be modifying the standard formula to exclude the bases. Tara will not be painting these parts of the post either. SA_(prism) (including the Base)=& Ph +2B SA_(prism)(excluding the Base)=& Ph Notice that the bases of the prism and pyramid are the same. We already calculated the perimeter of the base before. It's 24 inches. We are also given that the height of the prism is 4 feet. Because this value is measured in feet, while the rest of the given values are measured in inches, we should convert it into inches. 12inches/1foot By multiplying this conversion factor by our measurement, we can convert 4 feet into inches.

12inches/1foot* 4 feet

▼

Simplify

MoveRightFacToNum

a/c* b = a* b/c

12inches* 4 feet/1 foot

CrossCommonFac

Cross out common factors

12inches* 4 feet/1 foot

SimpTerms

Simplify terms

12* 4inches/1

Multiply

Multiply

48inches/1

DivByOne

a/1=a

48inches

Thus, the height of the prism is 48 inches. Now, we know both the perimeter and height of the pyramid. Let's substitute their values into the above formula and calculate the surface area of the prism faces.

SA_(prism)=Ph

SubstituteII

P= 24, h= 48

SA_(prism)= 24* 48

Multiply

Multiply

SA_(prism)=1152

The surface area of the prism's faces is 1152in^2.

Total Paintable Surface Area of the Post

So far we calculated the following surface areas. SA_(pyramid)=60 in^2 SA_(prism)=1152 in^2 Adding all these surfaces areas we get the total surface area of the post. SA_(post)=60+1152=1212in^2 The surface are of the fence post that is going to be painted is 1212in^2.

Volume

Since the post has quite a complicated shape, we can calculate its volume by adding the volumes of the pyramid and prism. Let's find them!

Pyramid's Volume

To calculate the volume of a pyramid we can use the formula V_(pyramid)= 1/3Bh where B is the area of the pyramid's base and h is the height. It is given that the height of the pyramid is 4 in. The base of the post is a square with the side of 6 inches. To calculate the area of a square we can use the following formula. B=a^2 Let's substitute a with 6. B= 6^2=36in^2 Therefore, the area of the base is 36in^2. Now, we can substitute the values of B and h into the formula and calculate V_(pyramid).

V_(pyramid)=1/3Bh

SubstituteII

B= 36, h= 4

V_(pyramid)=1/3* 36* 4

▼

Multiply

Multiply

Multiply

V_(pyramid)=1/3* 144

MoveRightFacToNumOne

1/b* a = a/b

V_(pyramid)=144/3

CalcQuot

Calculate quotient

V_(pyramid)=48

The volume of the pyramid is 48in^3.

Prism's Volume

The volume of a prism can be calculated by the formula V_(prism)=Bh where B is the area of the base and h is the height. The base's area of the prism is the same as the base's area of the pyramid, which is 36in^2. The height of the prism is 4 feet or 48 inches. Thereby, we know everything we need to calculate the volume of the prism.

V_(prism)=Bh

SubstituteII

B= 36, h= 48

V_(prism)= 36* 48

Multiply

Multiply

V_(prism)=1728

Hence, the volume of the prism is 1728in^3.

Total Volume

Let's gather the information we found. V_(pyramid)=48in^3 V_(prism)=1728in^3 If we add these volumes we get the volume of the fence post. V_(post)=48+1728=1776in^3 The total volume of the post is 1776in^3.