# Warm Up Write down objective and homework in agenda Lay out homework (Distance & PT worksheet) Homework (Volume worksheet) Get a calculator!!!

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Warm Up Write down objective and homework in agenda Lay out homework (Distance & PT worksheet) Homework (Volume worksheet) Get a calculator!!!

Unit 1 Common Core Standards 8.EE.7 Solve linear equations in one variable. a. Give examples of linear equations in one variable with one solution, infinitely many solutions, or no solutions. Show which of these possibilities is the case by successively transforming the given equation into simpler forms, until an equivalent equation of the form x = a, a = a, or a = b results (where a and b are different numbers). b. Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms. 8.G.6 Explain a proof of the Pythagorean Theorem and its converse. 8.G.7 Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real- world and mathematical problems in two and three dimensions. 8.G.8 Apply the Pythagorean Theorem to find the distance between two points in a coordinate system. A-CED.1 Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. Note: At this level, focus on linear and exponential functions. A-CED.4 Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. For example, rearrange Ohm's law V = IR to highlight resistance R. Note: At this level, limit to formulas that are linear in the variable of interest, or to formulas involving squared or cubed variables.

Unit 1 Common Core Standards A-REI.1 Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method. A-REI.11 Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions. Note: At this level, focus on linear and exponential functions. A-REI.3 Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. A-SSE.1 Interpret expressions that represent a quantity in terms of its context. a. Interpret parts of an expression, such as terms, factors, and coefficients. b. Interpret complicated expressions by viewing one or more of their parts as a single entity. For example, interpret P(1+r)n as the product of P and a factor not depending on P. Note: At this level, limit to linear expressions, exponential expressions with integer exponents and quadratic expressions. G-GMD.1 Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri's principle, and informal limit arguments. Note: Informal limit arguments are not the intent at this level. G-GMD.3 Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.* Note: At this level, formulas for pyramids, cones and spheres will be given. G-GPE.7 Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula.

Unit 1 Common Core Standards N-Q.1 Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays. N-Q.3 Choose a level of accuracy appropriate to limitations on measurement when reporting quantities. N-RN.1 Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents. For example, we define 51/3 to be the cube root of 5 because we want (51/3)3 = 5(1/3)3 to hold, so (5 1/3)3 must equal 5. N-RN.2 Rewrite expressions involving radicals and rational exponents using the properties of exponents. Note: At this level, focus on fractional exponents with a numerator of 1. MP.1 Make sense of problems and persevere in solving them. MP.2 Reason abstractly and quantitatively. MP.4 Model with mathematics. MP.7 Look for and make use of structure.

Warm Up

Vocabulary Area of a circle Area of a circle is given by times the square of the radius Circumference Circumference is the perimeter of or distance around a circle given by times the diameter of the circle. cone a solid, 3-dimensional figure with one vertex and one circular base. Cylinder A solid, 3-dimensional figure with a curved side and two circular, congruent bases that are in parallel planes Sphere A three dimensional solid that is perfectly round, ex. A ball. volume The number of unit cubes or cubic units needed to fill the space inside a three- dimensional figure

Cavalieri’s Principle can be used to show that the volume of any pyramid or cone is equal to the volume of a particular pyramid of the same height with a triangular base of equal area. To do this, it is necessary to prove that all the cross sections have the same area. For instance, if a cone has base area 8π units2 and height h units, a triangular pyramid with the same volume can be constructed if its base has area 8π units2 and its height is h units. Furthermore, these need not be regular pyramids or right cones. This argument proves the following theorem.

If all six faces of a rectangular prism are squares, it is a cube. Remember! Height Triangular prism Rectangular prism Cylinder Base Height Base Height Base

VOLUME OF PRISMS AND CYLINDERS WordsNumbersFormula Prism: The volume V of a prism is the area of the base B times the height h. Cylinder: The volume of a cylinder is the area of the base B times the height h. B = 2(5) = 10 units 2 V = 10(3) = 30 units 3 B =  (2 2 ) = 4  units 2 V = (4  )(6) = 24   75.4 units 3 V = Bh = (  r 2 )h

Find the volume of each figure to the nearest tenth. Example A. A rectangular prism with base 2 cm by 5 cm and height 3 cm. = 30 cm 3 B = 2 5 = 10 cm 2 V = Bh = 10 3 Area of base Volume of a prism

Find the volume of the figure to the nearest tenth. B. 4 in. 12 in. = 192  602.9 in 3 B =  (4 2 ) = 16  in 2 V = Bh = 16  12 Example Area of base Volume of a cylinder

Find the volume of the figure to the nearest tenth. C. 5 ft 7 ft 6 ft V = Bh = 15 7 = 105 ft 3 B = 6 5 = 15 ft 2 1212 Example Area of base Volume of a prism

Find the volume of the figure to the nearest tenth. A. A rectangular prism with base 5 mm by 9 mm and height 6 mm. = 270 mm 3 B = 5 9 = 45 mm 2 V = Bh = 45 6 Area of base Volume of prism Try This

Find the volume of the figure to the nearest tenth. B. 8 cm 15 cm B =  (8 2 ) = 64  cm 2 = (64  )(15) = 960   3,014.4 cm 3 Try This Area of base Volume of a cylinder V = Bh

Find the volume of the figure to the nearest tenth. C. 10 ft 14 ft 12 ft = 60 ft 2 = 60(14) = 840 ft 3 Try This Area of base Volume of a prism B = 12 10 1212 V = Bh

A pyramid is named for the shape of its base. The base is a polygon, and all of the other faces are triangles. A cone has a circular base. The height of a pyramid or cone is measured from the highest point to the base along a perpendicular line.

VOLUME OF PYRAMIDS AND CONES (2 2 )

Example: Finding the Volume of Pyramids and Cones Find the volume of the figure. 1313 V = 14 6 V = 28 cm 3 A. V = Bh 1313 B = (4 7) = 14 cm 2 1212

Example 2: Finding the Volume of Pyramids and Cones 1313 V = 9 10 V = 30  94.2 in 3 B. V = Bh 1313 B = (3 2 ) = 9 in 2 Use 3.14 for . Find the volume of the figure.

Example 3: Finding the Volume of Pyramids and Cones 1313 V = 84 10 V = 280 m 3 C. V = Bh 1313 B = 14 6 = 84 m 2 Find the volume of the figure.

Try This: Example 1A 1313 V = 17.5 7 V  40.8 in 3 A. V = Bh 1313 B = (5 7) = 17.5 in 2 1212 Find the volume of the figure. 5 in. 7 in.

1313 V = 9 7 V = 21  65.9 m 3 B. V = Bh 1313 B = (3 2 ) = 9 m 2 Use 3.14 for . Find the volume of the figure. Try This: Example 1B 7 m 3 m

1313 V = 16 8 V  42.7 ft 3 C. V = Bh 1313 B = 4 4 = 16 ft 2 Find the volume of the figure. Try This: Example 1C 4 ft 8 ft

A sphere is the set of points in three dimensions that are a fixed distance from a given point, the center. A plane that intersects a sphere through its center divides the two halves or hemispheres. The edge of a hemisphere is a great circle.

The volume of a hemisphere is exactly halfway between the volume of a cone and a cylinder with the same radius r and height equal to r.

Example 1: Finding the Volume of a Sphere Find the volume of a sphere with radius 9 cm, both in terms of  and to the nearest tenth of a unit. = 972 cm 3  3,052.1 cm 3 Volume of a sphere Substitute 9 for r. 4343 V = r 3 = (9) 3 4343

Try This: Example 1 Find the volume of a sphere with radius 3 m, both in terms of  and to the nearest tenth of a unit. = 36 m 3  113.0 m 3 Volume of a sphere Substitute 3 for r. 4343 V = r 3 = (3) 3 4343

The surface area of a sphere is four times the area of a great circle.  50.3 units 2

Additional Example 2: Finding Surface Area of a Sphere Find the surface area, both in terms of  and to the nearest tenth of a unit. = 36 in 2  113.0 in 2 S = 4r 2 = 4(3 2 ) Surface area of a sphere Substitute 3 for r.

Try This: Example 2 The moon has a radius of 1738 km. Find the surface area, both in terms of  and to the nearest tenth. = 12,082,576 km 2  37,939,288.6 km 2 S = 4r 2 = 4(1738 2 ) Surface area of a sphere Substitute 1738 for r. 1738 km

Additional Example 3: Comparing Volumes and Surface Areas Sphere:  310,464 cm 3 Rectangular Prism: = (44)(84)(84) = 310,464 cm 3 V = lwh V = r 3 = (42 3 ) 4343 4343  74,088 22 7 4343 Compare the volumes and surface areas of a sphere with radius 42 cm with that of a rectangular prism measuring 44 cm  84 cm  84 cm.

S = 4r 2 = 4(42 2 ) = 7,056 = 28,896 cm 2 S = 2(44)(84) + 2(44)(84) + 2(84)(84) Additional Example 3 Continued Sphere: Rectangular Prism: S = 2lw + 2lh + 2wh  7,056  22,176 cm 2 22 7 The sphere and the prism have approximately the same volume, but the prism has a larger surface area. Compare the volumes and surface areas of a sphere with radius 42 cm with that of a rectangular prism measuring 44 cm  84 cm  84 cm.

Try This: Example 3 Sphere:  38,808 mm 3 Rectangular Prism: = (22)(42)(42) = 38,808 mm 3 V = lwh V = r 3 = (21 3 ) 4343 4343  9261 22 7 4343 Compare the volume and surface area of a sphere with radius 21 mm with that of a rectangular prism measuring 22  42  42 mm.

S = 4r 2 = 4(21 2 ) = 1764 = 7224 mm 2 S = 2(22)(42) + 2(22)(42) + 2(42)(42) Sphere: Rectangular Prism: S = 2lw + 2lh + 2wh  1764  5544 mm 2 22 7 The sphere and the prism have approximately the same volume, but the prism has a larger surface area. Try This: Example 3 Continued Compare the volume and surface area of a sphere with radius 21 mm with that of a rectangular prism measuring 22  42  42 mm.

Lesson Quiz: Part 1 Find the volume of each sphere, both in terms of  and to the nearest tenth. Use 3.14 for . 1. r = 4 ft 2. d = 6 m Find the surface area of each sphere, both in terms of  and to the nearest tenth. Use 3.14 for . 36m 3, 113.0 m 3 85.3ft 3, 267.8 ft 3 2.25 mi 2, 7.1 mi 2 1936 in 2, 6079.0 in 2 3. r = 22 in 4. d = 1.5 mi

Extra Practice Volume of a cylinder: http://regentsprep.org/Regents/math/geomet ry/GG2/CylinderPage.htm http://regentsprep.org/Regents/math/geomet ry/GG2/CylinderPage.htm Volume of a cone: http://regentsprep.org/Regents/math/geomet ry/GG2/ConePage.htm http://regentsprep.org/Regents/math/geomet ry/GG2/ConePage.htm Volume of a sphere: http://regentsprep.org/Regents/math/geomet ry/GG2/SpherePage.htm http://regentsprep.org/Regents/math/geomet ry/GG2/SpherePage.htm

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