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This page is going to explain how to develope the concepts of maximizing volume and minizing surface area.



Link to volume of sphere
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Surface Area and Volume Formulas


Objectives
-To explore the ratios of surface area to volume.
-To develope the concepts of maximizing volume and minimizing surface area.


The surface area (S) and Volume (V) of a right rectangular prism with length ( l) width (w) and height (h) are
S= 2lh + 2wh + 2lh and V= lwh
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The surface (S) area and volume (V) of a cube with side (s) are
S= 6s² and V= s³

Example:
A shipping company is choosing between two box designs. (Box A and Box B)
Which design has the greater surface area and requires more material for the same volume?

Both box's have a volume of 160 cubic inches.
SA of box A is: 2(8) (5 ) + 2(10) (8) = 184 square inches.
SA of box B is: 2(10) (8) + 2(2) (10) = 232 square inches.
Answer: Box B has the greater surface area.



Surface Area and Volume of Prisms


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Objectives:
-Define and use a formula for finding the surface area of a right prism.
-Define and use a formula for finding the volume or a right prism.
-Use Cavalieri's Principle to develope a formula for the volume of a right or oblique prism.

An altitude of a prism is a segment hat has endpoints in the planes containing the bases and that is perpendicular to both planes.
The height of a prism is the length of an altitude.


Surface Area of Right Prisms


The surface area of a prism may be broken down into two parts: the area of the bases, or base area, and the area of the lateral faces, or lateral area.
Since the bases are congruent, the base area is twice the area of one base, or 2B, where B is the area of one base.
If the sides of the base are s¹, s², and s³ and the height is h, then the lateral area is given by the following formula:
L = s¹h + s²h + s³h = h(s¹ + s² + s³)
Because s¹ + s² + s³ is the perimeter of the base, we can write the lateral area as L = hp, where p is the perimeter of the base.
- The surface area of a prism is the sum of the base area and the lateral area.

The surface area, S, of a right prism with lateral area L, base area B, perimeter p, and height h is :
S = L + 2 B or S = hp + 2B

Example 1:
The area of each base is
B = ½ (2) (21) = 21.
The perimeter of each base is
p = 10 + 21 + 17 = 48,
so the lateral area is
L = hp = 30 (48) = 1440.

So then the surface area is
S = L + 2B = 1440 + 2 (21) = 1440 + 42 = 1482.

Volumes of Right Prisms

The volume of a right rectangular prism with length L, width w, and height h, is given by V = lwh. Because the base area, B, of this type of prism is equal to lw, you can also write the formula for the volume as V = Bh.

Example 2:

An aquarium in the shape of a right rectangular prism has dimensions of 110 × 50 × 7 feet.
Given that 1 gallon is approximately 0.134 cubic feet, how many gallons of water will the aquarium hold?
Given 1 gallon of water is approximately 8.33 pounds, how much will the water weigh?

Answer:
The volume of the aquarium is found by using the volume formula.
V = Bh = lwh = (110) (50) (7) = 38,500 cubic feet.
To approximate the volume in gallons, divide by 0.134.
V = 38,500 ÷ 0.134 aprrox 287,313 gallons
To approximate the weight, multiply by 8.33.
weight approx (287,313) (8.33) approx 2,393,317 pounds.

Example 3:

An aquarium has the shape of a right regular hexagonal prism.

The base of the aquarium has a perimeter of (14) (6), or 84, inches and an apothem of 7÷3 inches, so the base area is found as follows:
B = ½ ap = ½ (84) (7 ÷ 3) = 294 ÷ 3 approx 509.22 square inches
The volume is V = Bh = (294 ÷ 3) (48) = 14112 ÷ 3 approx 24,443 cubic inches.

Volumes of Oblique Prisms


In an oblique prism, the lateral edges are not perpendicular to the bases, and there is no simple general formula for surface area. Although, the formula for the volume is the same as that for a right prism. To understand why this is true, consider the explanation below.
Stack a set of index cards in the shape of a right rectangular prism. If you push the stack into the shape of an oblique prism, the volume of the solid does not change because the number of cards does not change.
Both stacks have the same number of cards, and each prism is the same height.
Because every card has the same size and shape, they all have the same area. Any card in either stack represents a cross section of each prism.
Cavalieri's Principle
If two solids have equal heights and the cross sections formed by every plane parrallel to the bases of both solid have equal areas, then the two solids have equal volumes.

Volume of a Prism
The Volume, V, of a prism with height h and base area B is
V = Bh.

Surface Area and Volume of Pyramids


Objectives:
-Define and use a formula for the surface area of a regular pyramid.
-Define and use a formula for the volume of a pyramid.
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A pyramid is a polyhedron consisting of a base, which is a polygon, and three or more lateral faces.
The lateral faces are triangles that sharea a single vertex, called the vertex of the pyramid.
Each lateral face has one edge in common with the base, called a base edge.
The intersection of two lateral faces is a lateral edge.

The altitude of a pyramid is the perpendicular segment from the vertex to the plane of the base. The height of a pyramid is the length of its altitude.

A regular pyramid is a pyramid whose base is a regular polygon and whose lateral faces are congruent isosceles triangles. In a regular pyramid, all of the lateral edges area congruent, and the altitude intersects the base at its center. The length of an altitude of a lateral face of a regular pyramid is called the slant height of the pyramid.
Pyramids, like prisms, are named by the shape of their base.

Surface Area of a Pyramid

Example 1:
Find the surface area of a regular square pyramid whose slant height is L and whose base edge length is s.
Answer:

The surface area is the sum of the lateral areas and the base area.
S = L + B
S = 4 (½sl) + s²
This can be rewritten as follows:
S = ½Lp + s².

Surface Area of a Regular Pyramid

The surface area, S, of a regular pyramid with lateral area L, base area B, perimeter of the base p, and slant height L is
S = L + B or S = ½lp + B

Example 2:
The roof of a gazebo is a regular octagonal pyramid with a base edge of 4 feet and a slant height of 6 feet.
Find the area of the roof. If roofing material costs $3.50 per square foot, find the cost of covering the roof with this material.gazebo.jpg
Answer:
The area of the roof is the lateral area of the pyramid.
L = ½lp = ½ (6) (8 × 4) = 96 square feet
96 square feet × $3.50 per square foot = $336.00


Volume of a Pyramid
The volume,V, of a pyramid with height h and base area B is
V = 1/3bh
Example 3:
The pyramid of Khufu is a regular square pyramid with a base edge of approxiamtely
776 feet and an original height of 481 feet. The limestone used to construct the pyramid weighs
approxmately 167 pounds per cubic foot. Estimate the weight of the pyramid of Khufu. (Assume the pyramid is solid)
Answer:
The volume of the pyramid is found as follows:
V = 1/3Bh
approx 1/3 (776²) ( 481)
approx 96,548,885 cubic feet
The weight in pounds is 96,548,885 cubic feet × 167 pounds per cubic foot approx 16,123,663,850
pounds, or 8,061,831 tons.

Surface Area and Volume of Cylinders

Objectives:
-Define and use a formula for the surface area of a right cylinder.
-Define and use a formula for the volume of a cylinder.
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Cylinders
A cylinder is a solid that consists of a circular region and its translated
image on a parallel plane, with lateral surface connecting the circles.

The faces formed by the circular region and its translated image are called
the bases of the cylinder.

An altitude of a cylinder is a segment that has endpoints in the planes containing the bases and is perpendicular to both planes. The height of a
cylinder is the length of an altitude.

Th axis of a cylinder is the segment joining the centers of the two bases.

If the axis of a cylinder is perpendicular to the bases, then the cylinder is right cylinder.
If not, it is an oblique cylinder.

Cylinders and Prisms

As the number of lateral faces of a regular polygonal prism increases, the figure becomes more and more
like a circle.
Similarity, as the number of lateral faces of a regular polygonal prism incresses, the figure becomes more and more like a cylinder.
This fact suggests that the formulas for surface areas and volumes of prisms and cylinders are similar.

Surface Area of a Right Cylinder
The surface area,S, of a right cylinder with lateral area L, base area B, radius r, and height h is
S= L + 2B or S = 2pierh + 2pier ²
Example 1:
A pennypenny.jpg is a right cylinder with a diameter of 19.05 millimeters
and a thickness of 1.55 millimeters. Ignoring the raised design, estimate
the surface area of a penny.
Answer:
The radius of a penny is half of the diameter, or 9.525 millimeters. Use
the formula for the sirface area of a right cylinder.
S = 2pie rh + 2pie r ²
S = 2pie (9.525) (1.55) + 2pie (9.525)² approx 663.46 square
millimeters.

Volume of a Cylinder

The volume,V, of a cylinder with radius r, height h, and base area B is
V = Bh or V = pie r²h.
Example 2:
A tank has a length of 31 feet 6½ inches and an outer diameter of 8 feet 0 inches. Assuming a wall thickness of about 2 inches, what is the volume of the tank? At 15 gallons per car, how many car tanks could be filled from the storage tank if it starts out completely full of gasoline?Tanks.jpg
Answer:
The tank is not perfectly cylindrical, because of its hemispherical heads, but you can approximate its volume by a slightly shorter cylindrical tank, say 29 feet long. Subtracting the wall thickness from the dimensions of the tank,
V = pie r² h approx pie (3.833)² (28.667) approx 1323 cubic feet
Convert from cubic feet to gallons.
1323 cubic feet × 7.48 gallons per cubic foot approx 9896 gallons
So the tank could deliver about 9896/15, or approx 660,15-gallon fill-ups.

Surface Area and Volume of Cones

Objectives:
- Define and use the fomulas for the surface area of a cone.
- Define and use the formula for the volume of a cone.
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Cones
A cone is a three-dimenstional figure that consists of a circular base and a curved lateral surface that connects the base to a single point not in the plane of the base, called the vertex.

The altitude of a cone is the perpendicular segment from the vertex to the plane of the base. The height of the cone is the length of the altitude.

If the altitude of a cone intersects the base of the cone at its center, the cone is a right cone. If not, it is an oblique cone.

Surface Area of a right cone
Example 1:
Find the surface area of a right cone with the indicated measurements.
Answer:
The circumfrence of the base is c = 2pie r = 14pie.
The lateral area is a sector of a circular region with curcumference C = 2pieL = 30pie.
The portion of the circular region occupied by the sector is c/C = 14pie/30pie = 7/15.
Calculate the area of the sector ( lateral area).
pieL² = 225pie
L = 7/15 • 225pie = 105pie
Calculate the base area and add the lateral area.
B = pier² = 49pie
B +