Area and Volume Equations

Transcription

1 Area and Volume Equations MODULE 16? ESSENTIAL QUESTION How can you use area and volume equations to solve real-world problems? LESSON 16.1 Area of Quadrilaterals 6.8.B, 6.8.D LESSON 16. Area of Triangles 6.8.B, 6.8.D LESSON 16.3 Solving Area Equations 6.8.C, 6.8.D LESSON 16.4 Solving Volume Equations 6.8.C, 6.8.D Image Credits: Getty Royalty Free Real-World Video Quilting, painting, and other art forms use familiar geometric shapes, such as triangles and rectangles. To buy enough supplies for a project, you need to find or estimate the areas of each shape in the project. Math On the Spot Animated Math Go digital with your write-in student edition, accessible on any device. Scan with your smart phone to jump directly to the online edition, video tutor, and more. Interactively explore key concepts to see how math works. Get immediate feedback and help as you work through practice sets. 443

2 Are YOU Ready? Complete these exercises to review skills you will need for this chapter. Use of Parentheses Online Assessment and Intervention EXAMPLE 1_ (14) (1 + 18) = 1_ (14) (30) Perform operations inside parentheses first. = 7 (30) = 10 Multiply left to right. Multiply again. Evaluate. 1. 1_ (3) (5 + 7). 1_ (15) ( ) 3. 1_ (10) ( ) 4. 1_ (.1) ( ) Area of Square, Rectangles, Triangle EXAMPLE Find the area of the rectangle. 8 ft 3 ft A = bh Use the formula for area of a rectangle. = 8 3 Substitute for base and height. = 4 Multiply. Area equals 4 square feet. Find the area of each figure. 5. a triangle with base 6 in. and height 3 in. 6. a square with sides of 7.6 m 7. a rectangle with length 3 1_ 4 ft and width 1_ ft 8. a triangle with base 8. cm and height 5.1 cm 444 Unit 5

3 Reading Start-Up Visualize Vocabulary Use the words to complete the graphic. You will put one word in each oval. Types of Triangles Vocabulary Review Words base (base) equilateral triangle (triángulo equilátero) isosceles triangle (triángulo isósceles) legs (catetos) quadrilateral (cuadrilátero) rectangular prism (prisma rectangular) right triangle (triángulo rectángulo) volume (volumen) has three congruent sides and three congruent angles has two equal sides and two equal angles contains a 90 angle Preview Words parallelogram (paralelogramo) rhombus (rombo) trapezoid (trapecio) Understand Vocabulary Match the term on the left to the correct expression on the right. 1. parallelogram A. A quadrilateral in which all sides are congruent and opposite sides are parallel.. trapezoid B. A quadrilateral in which opposite sides are parallel and congruent. 3. rhombus C. A quadrilateral in which two sides are parallel. Active Reading Booklet Before beginning the module, create a booklet to help you learn the concepts in this module. Write the main idea of each lesson on each page of the booklet. As you study each lesson, write important details that support the main idea, such as vocabulary and formulas. Refer to your finished booklet as you work on assignments and study for tests. Module

4 MODULE 16 Unpacking the TEKS Understanding the TEKS and the vocabulary terms in the TEKS will help you know exactly what you are expected to learn in this module. 6.8.C Write equations that represent problems related to the area of rectangles, parallelograms, trapezoids, and triangles and volume of right rectangular prisms where dimensions are positive rational numbers. What It Means to You You will use the formula for the area of a figure to write an equation that can be used to solve a problem. UNPACKING EXAMPLE 6.8.C The Hudson Middle School wrestling team won the state tournament and was awarded a triangular pennant to display in the school gymnasium. The pennant has an area of.5 square meters. The base of the pennant is 1.5 meters long. Write an equation to find the height of the pennant. A = 1_ bh.5 = 1_ ( 1.5)h.5 = 0.75h 1.5 m An equation to find the height of the pennant is.5 = 0.75h. 6.8.D Determine solutions for problems involving the area of rectangles, parallelograms, trapezoids, and triangles and volume of right rectangular prisms where dimensions are positive rational numbers. Visit to see all the unpacked. What It Means to You You will use the formula for the volume of a rectangular prism. UNPACKING EXAMPLE 6.8.D Jala has an aquarium in the shape of a rectangular prism with a volume of,160 cubic inches. The length is 15 inches and the width is 1 inches. Find the height of the aquarium. v = l w h,160 = 15 1 h,160 = 180 h, = h 1 = h The height of the aquarium is 1 inches. Image Credits: GK Hart/Vikki Hart/Getty Images 446 Unit 5

5 ? LESSON 16.1 ESSENTIAL QUESTION Area of Quadrilaterals How can you find the areas of parallelograms, rhombuses, and trapezoids? Expressions, equations, and relationships 6.8.B Model area formulas for parallelograms, trapezoids... by decomposing and rearranging parts of these shapes. Also 6.8.D EXPLORE ACTIVITY 6.8.B Area of a Parallelogram Recall that a rectangle is a special type of parallelogram. A B Draw a large parallelogram on grid paper. Cut out your parallelogram. Cut your parallelogram on the dashed line as shown. Then move the triangular piece to the other side of the parallelogram. height (h) width (w) base (b) length (l) C What figure have you formed? Does this figure have the same area as the parallelogram? base of parallelogram = of rectangle D height of parallelogram = of rectangle area of parallelogram = of rectangle What is the formula for the area of this figure? A = or What is the formula for the area of a parallelogram? A = Area of a Parallelogram Math Talk Mathematical Processes How is the relationship between the length and width of a rectangle similar to the relationship between the base and height of a parallelogram? The area A of a parallelogram is the product of its base b and its height h. A = bh h b Lesson

6 EXPLORE ACTIVITY (cont d) Reflect 1. Find the area of the parallelogram. A = 7 cm 14 cm Math On the Spot Finding the Area of a Trapezoid To find the formula for the area of a trapezoid, notice that two copies of the same trapezoid fit together to form a parallelogram. Therefore, the area of the trapezoid is 1_ the area of the parallelogram. h The height of the parallelogram is the same as the height of the trapezoid. The base of the parallelogram is the sum of the two bases of the trapezoid. b A = b h A = ( b 1 + b ) h Area of a Trapezoid The area of a trapezoid is half its height multiplied by the sum of the lengths of its two bases. A = 1 h( b 1 + b ) h b 1 b Math Talk Mathematical Processes Does it matter which of the trapezoid s bases is substituted for b 1 and which is substituted for b? Why or why not? EXAMPLE 1 A section of a deck is in the shape of a trapezoid. What is the area of this section of the deck? 17 ft b 1 = 17 b = 39 h = 16 Use the formula for area of a trapezoid. A = 1 h( b 1 + b ) = 1 16( ) Substitute. 16 ft 39 ft 6.8.D Animated Math = 1 16(56) = 8 56 = 448 square feet Add inside the parentheses. Multiply 1 and 16. Multiply. 448 Unit 5

7 YOUR TURN. Another section of the deck is also shaped like a trapezoid. For this section, the length of one base is 7 feet, and the length of the other base is 34 feet. The height is 1 feet. What is the area of this section of the deck? A = ft Online Assessment and Intervention Finding the Area of a Rhombus A rhombus is a quadrilateral in which all sides are congruent and opposite sides are parallel. A rhombus can be divided into four triangles that can then be rearranged into a rectangle. Math On the Spot h b The base of the rectangle is the same length as one of the diagonals of the rhombus. The height of the rectangle is 1_ the length of the other diagonal. A = b h A = d 1 1_ d Area of a Rhombus The area of a rhombus is half of the product of its two diagonals. A = 1_ d 1 d d d 1 EXAMPLE Cedric is constructing a kite in the shape of a rhombus. The spars of the kite measure 15 inches and 4 inches. How much fabric will Cedric need for the kite? To determine the amount of fabric needed, find the area of the kite. d 1 = 15 d = 4 Use the formula for area of a rhombus. 6.8.B 15 in. A = 1_ d 1 d = 1_ (15)(4) = 180 square inches Substitute. Multiply. 4 in. Lesson

8 YOUR TURN Online Assessment and Intervention Find the area of each rhombus. 3. d 4. 1 = 35 m; d = 1 m A = m 5. d 1 = 10 m; d 6. = 18 m A = m d 1 = 9.5 in.; d = 14 in. A = in d1 = 8 1_ 4 ft; d = 40 ft A = ft Guided Practice 1. Find the area of the parallelogram. (Explore Activity) A = bh = ( )( ) = in.. Find the area of the trapezoid. (Example 1) A = 1_ h( b 1 + b ) = 1_ ( )( + ) 9 in. 13 in. 9 cm 14 cm = cm 3. Find the area of the rhombus. (Example ) 15 cm? A = 1_ d 1 d = 1_ ( )( ) = in. ESSENTIAL QUESTION CHECK-IN 18 in. 11 in. 4. How can you find the areas of parallelograms, rhombuses, and trapezoids? 450 Unit 5

9 Name Class Date 16.1 Independent Practice 6.8.B, 6.8.D 5. Rearrange the parts of the parallelogram to form a rectangle. Find the area of the parallelogram and the area of the rectangle. What is the relationship between the areas? Online Assessment and Intervention 9. The seat of a bench is in the shape of a trapezoid with bases of 6 feet and 5 feet and a height of 1.5 feet. What is the area of the seat? 6 cm 14 cm 10. A kite in the shape of a rhombus has diagonals that are 5 inches long and 15 inches long. What is the area of the kite? 6. What is the area of a parallelogram that has a base of 1 3 _ 4 in. and a height of 1_ in.? 11. A window in the shape of a parallelogram has a base of 36 inches and a height of 45 inches. What is the area of the window? 7. Draw a copy of the trapezoid to form a parallelogram. Find the area of the trapezoid and the area of the parallelogram. What is the relationship between the areas? 4 in. 1. Communicate Mathematical Ideas Find the area of the figure. Explain how you found your answer. 6 ft 10 ft 4 in. 36 in. 18 ft 1 ft 8. The bases of a trapezoid are 11 meters and 14 meters. Its height is 10 meters. What is the area of the trapezoid? Lesson

10 13. Multistep A parking space shaped like a parallelogram has a base of 17 feet and a height is 9 feet. A car parked in the space is 16 feet long and 6 feet wide. How much of the parking space is not covered by the car? FOCUS ON HIGHER ORDER THINKING Work Area 14. Critique Reasoning Simon says that to find the area of a trapezoid, you can multiply the height by the top base and the height by the bottom base. Then add the two products together and divide the sum by. Is Simon correct? Explain your answer. 15. Multistep The height of a trapezoid is 8 in. and its area is 96 in. One base of the trapezoid is 6 inches longer than the other base. What are the lengths of the bases? Explain how you found your answer. 16. Critique Reasoning Find the area of the trapezoid using the formula A = 1_ h(b 1 + b ). Decompose the trapezoid into a rectangle and a triangle and find the area of each. Then find the sum of the two areas. Compare this sum with the area of the trapezoid. 8 cm 1 cm 6 cm 45 Unit 5

11 ? LESSON 16. Area of Triangles ESSENTIAL QUESTION How do you find the area of a triangle? Expressions, equations, and relationships 6.8.B Model area formulas for... triangles by decomposing and rearranging parts of these shapes. Also 6.8.D EXPLORE ACTIVITY 1 Area of a Right Triangle A Draw a large rectangle on grid paper. 6.8.B b h What is the formula for the area of a rectangle? A = B Draw one diagonal of your rectangle. The diagonal divides the rectangle into. Each one represents of the rectangle. Use this information and the formula for area of a rectangle to write a formula for the area of a right triangle. A = Reflect 1. Communicate Mathematical Ideas In the formula for the area of a right triangle, what do b and h represent? EXPLORE ACTIVITY Area of a Triangle A B C 6.8.B Draw a large triangle on grid paper. Do not draw a right triangle. Cut out your triangle. Then trace around it to make a copy of your triangle. Cut out the copy. Cut one of your triangles into two pieces by cutting through one angle directly across to the opposite side. Now you have three triangles one large triangle and two smaller triangles. Lesson

12 EXPLORE ACTIVITY (cont d) When added together, the areas of the two smaller triangles D equal the of the large triangle. Arrange the three triangles into a rectangle. What fraction of the rectangle does the large triangle represent? h The area of the rectangle is A = bh. What is the area of the large triangle? A = b How does this formula compare to the formula for the area of a right triangle that you found in Explore Activity 1? Reflect. Communicate Mathematical Ideas What type of angle is formed by the base and height of a triangle? Finding the Area of a Triangle Area of a Triangle Math On the Spot EXAMPLE 1 Find the area of each triangle. A The area A of a triangle is half the product of its base b and its height h. A = 1_ bh b = 0 meters b h h = 8 meters 6.8.D 0 m 8 m A = 1_ bh = 1_ (0 meters) (8 meters) = 80 square meters Substitute. Multiply. 454 Unit 5

13 Find the area of each triangle. B 5 in. 1 in. b = 1 inches A = 1_ bh h = 5 inches = 1_ (1 inches) (5 inches) = 30 square inches Substitute. Multiply. YOUR TURN Find the area of the triangle in. Online Assessment and Intervention 8.5 in. A = Math Talk Mathematical Processes Why can you also write the formula for the area of a triangle as A = bh? Image Credits: JOHN MACDOUGALL/AFP/Getty Images Problem Solving Using Area of Triangles You can use the formula for the area of a triangle to solve real-world problems. EXAMPLE Each triangular face of the Pyramid of Peace in Kazakhstan is made up of 5 smaller equilateral triangles. These triangles have measurements as shown in the diagram. What is the area of one of the smaller equilateral triangles? STEP 1 STEP Identify the length of the base and the height of the triangle. b = 1 m and h = 10.4 m Use the formula to find the area of the triangle. A = 1_ bh Substitute. = 1_ (1)(10.4) Multiply. = m 6.8.D 10.4 m Math On the Spot The area of one small equilateral triangle is 6.4 m. Lesson

14 YOUR TURN Online Assessment and Intervention 4. Amy needs to order a shade for a triangular-shaped window that has a base of 6 feet and a height of 4 feet. What is the area of the shade? Guided Practice 1. Show how you can use a copy of the triangle to form a rectangle. Find the area of the triangle and the area of the rectangle. What is the relationship between the areas? (Explore Activities 1 and, Example 1) 8 in. 14 in. Triangle: A = 1_ bh = 1_ ( ) ( ) = in Rectangle: A = bh = ( ) ( ) = in?. A pennant in the shape of a triangle has a base of 1 inches and a height of 30 inches. What is the area of the pennant? (Example ) A = 1_ bh = 1_ ( ) ( ) = in ESSENTIAL QUESTION CHECK-IN 3. How do you find the area of a triangle? 456 Unit 5

15 Name Class Date 16. Independent Practice 6.8.B, 6.8.D Find the area of each triangle Online Assessment and Intervention 10 cm 0 ft 15 cm 4 ft in. 18 ft 3 ft 1 in. 8. What is the area of a triangle that has a 9. base of 15 1_ 4 in. and a height of 18 in.? A right triangle has legs that are 11 in. and 13 in. long. What is the area of the triangle? 10. A triangular plot of land has the dimensions shown in the diagram. What is the area of the land? 11. The front part of a tent has the dimensions shown in the diagram. What is the area of this part of the tent? 30 km 0 km 1. Multistep The sixth-grade art students are making a mosaic using tiles in the shape of right triangles. Each tile has leg measures of 3 centimeters and 5 centimeters. If there are 00 tiles in the mosaic, what is the area of the mosaic? 5 ft 8 ft 13. Critique Reasoning Monica has a triangular piece of fabric. The height of the triangle is 15 inches and the triangle s base is 6 inches. Monica says that the area of the fabric is 90 in. What error did Monica make? Explain your answer. Lesson

16 14. Show how you can use the given triangle and its two smaller right triangles to form a rectangle. What is the relationship between the area of the original triangle and the area of the rectangle? 1 in. 18 in. FOCUS ON HIGHER ORDER THINKING Work Area 15. Communicate Mathematical Ideas Explain how the areas of a triangle and a parallelogram with the same base and height are related. 16. Analyze Relationships A rectangle and a triangle have the same area. If their bases are the same lengths, how do their heights compare? Justify your answer. 17. What If? A right triangle has an area of 18 square inches. a. If the triangle is an isosceles triangle, what are the lengths of the legs of the triangle? b. If the triangle is not an isosceles triangle, what are all the possible lengths of the legs, if the lengths are whole numbers? 458 Unit 5

17 ? LESSON 16.3 ESSENTIAL QUESTION Solving Area Equations Expressions, equations, and relationships 6.8.C Write equations that represent problems related to the area of rectangles, parallelograms, trapezoids, and triangles where dimensions are positive rational numbers. Also 6.8.D How do you use equations to solve problems about area of rectangles, parallelograms, trapezoids, and triangles? Problem Solving Using the Area of a Triangle Recall that the formula for the area of a triangle is A = 1_ bh. You can also use the formula to find missing dimensions if you know the area and one dimension. EXAMPLE D Math On the Spot The Hudson High School wrestling team just won the state tournament and has been awarded a triangular pennant to hang on the wall in the school gymnasium. The base of the pennant is 1.5 feet long. It has an area of.5 square feet. What is the height of the pennant? 1.5 ft A = 1_ bh.5 = 1_ (1.5)h.5 = 0.75h = h 3 = h Write the formula. The height of the pennant is 3 feet. YOUR TURN Use the formula to write an equation. Multiply 1 and 1.5. Divide both sides of the equation by Renee is sewing a quilt whose pattern contains right triangles. Each quilt piece has a height of 6 in. and an area of 4 in. How long is the base of each quilt piece? Math Talk Mathematical Processes How can you use units in the formula to confirm that the units for the height are in feet? Online Assessment and Intervention Lesson

18 Math On the Spot Writing Equations Using the Area of a Trapezoid You can use the formula for area of a trapezoid to write an equation to solve a problem. EXAMPLE A garden in the shape of a trapezoid has an area of 44.4 square meters. One base is 4.3 meters and the other base is 10.5 meters long. The height of the trapezoid is the width of the garden. How wide is the garden? A = 1_ h (b 1 + b ) 44.4 = 1_ h ( ) Write the formula. 4.3 m Use the formula to write an equation m 6.8.C Math Talk Mathematical Processes How can you check that the answer is reasonable? 44.4 = 1_ h (14.8) 44.4 = 7.4 h = h 6 = h The garden is 6 meters wide. Add inside parentheses. Multiply 1 and Divide both sides of the equation by 7.4. Reflect. Communicate Mathematical Ideas Explain why the first step after substituting is addition. Online Assessment and Intervention YOUR TURN 3. The cross section of a water bin is shaped like a trapezoid. The bases of the trapezoid are 18 feet and 8 feet long. It has an area of 5 square feet. What is the height of the cross section? 460 Unit 5

19 Solving Multistep Problems You can write equations to solve real-world problems using relationships in geometry. EXAMPLE 3 Problem Solving 6.8.D Math On the Spot John and Mary are using rolls of fabric to make a rectangular stage curtain for their class play. The rectangular piece of fabric on each roll measures.5 feet by 15 feet. If the area of the curtain is 00 square feet, what is the least number of rolls they need? Analyze Information Rewrite the question as a statement. Find the least number of rolls of fabric needed to cover an area of 00 ft. List the important information. Each roll of fabric is a.5 foot by 15 foot rectangle. The area of the curtain is 00 square feet. Formulate a Plan Write an equation to find the area of each roll of fabric. Use the area of the curtain and the area of each roll to write an equation to find the least number of rolls. Solve STEP 1 STEP STEP 3 Write an equation to find the area of each roll of fabric. A = lw A = 15.5 A = 37.5 ft Write an equation to find the least number of rolls. n = n = 5 1_ 3 The problem asks for the least number of rolls needed. Since 5 rolls will not be enough, they will need 6 rolls to make the curtain. John and Mary will need 6 rolls of fabric to make the curtain. Justify and Evaluate The area of each roll is about 38 ft. Since 38 ft 6 = 8 ft, the answer is reasonable. Lesson

20 YOUR TURN Online Assessment and Intervention 4. A parallelogram-shaped field in a park needs sod. The parallelogram has a base of 1.5 meters and a height of 18 meters. The sod is sold in pallets of 50 square meters. How many pallets of sod are needed to fill the field? Guided Practice 1. A triangular bandana has an area of 70 square inches. The height of the triangle is 8 3_ 4 inches. Write and solve an equation to find the length of the base of the triangle. (Example 1). The top of a desk is shaped like a trapezoid. The bases of the trapezoid are 6.5 and 30 centimeters long. The area of the desk is 791 square centimeters. The height of the trapezoid is the width of the desk. Write and solve an equation to find the width of the desk. (Example ) 3. Taylor wants to paint his rectangular deck that is 4 feet long and 8 feet wide. A gallon of paint covers about 350 square feet. How many gallons of paint will Taylor need to cover the entire deck? (Example 3) Write an equation to find the of the deck. Write and solve the equation.? Write an equation to find the. Write and solve the equation. Taylor will need gallons of paint. ESSENTIAL QUESTION CHECK-IN 4. How do you use equations to solve problems about area of rectangles, parallelograms, trapezoids, and triangles? 46 Unit 5

21 Name Class Date 16.3 Independent Practice 6.8.C, 6.8.D Online Assessment and Intervention 5. A window shaped like a parallelogram has an area of 18 1_ square 3 feet. The height of the window is 3 1_ feet. How long is the base of 3 the window? 6. A triangular sail has a base length of.5 meters. The area of the sail is 3.75 square meters. How tall is the sail? 7. A section in a stained glass window is shaped like a trapezoid. The top base is 4 centimeters and the bottom base is.5 centimeters long. If the area of the section of glass is 3.9 square centimeters, how tall is the section? 8. Multistep Amelia wants to paint three walls in her family room. Two walls are 6 feet long by 9 feet wide. The other wall is 18 feet long by 9 feet wide. a. What is the total area of the walls that Amelia wants to paint? b. Each gallon of paint covers about 50 square feet. How many gallons of paint should Amelia buy to paint the walls? 9. Critical Thinking The area of a triangular block is 64 square inches. If the base of the triangle is twice the height, how long are the base and the height of the triangle? 10. Multistep Alex needs to varnish the top and the bottom of a dozen rectangular wooden planks. The planks are 8 feet long and 3 feet wide. Each pint of varnish covers about 15 square feet and costs $3.50. a. What is the total area that Alex needs to varnish? b. How much will it cost Alex to varnish all the wooden planks? 11. Multistep Leia cuts congruent triangular patches with an area of 45 square centimeters from a rectangular piece of fabric that is 18 centimeters long and 10 centimeters wide. How many of the patches can Leia cut from 3 pieces of the fabric? 1. Multistep A farmer needs to buy fertilizer for two fields. One field is shaped like a trapezoid, and the other is shaped like a triangle. The trapezoidal field has bases that are 35 and 48 yards and a height of 6 yards. The triangular field has the same height and a base of 39 yards. Each bag of fertilizer covers 150 square yards. Use a problem solving model to find how many bags of fertilizer the farmer needs to buy. Lesson

22 13. A tennis court for singles play is 78 feet long and 7 feet wide. a. The court for doubles play has the same length but is 9 feet wider than the court for singles play. How much more area is covered by the tennis court used for doubles play? b. The junior court for players 8 and under is 36 feet long and 18 feet wide. How much more area is covered by the tennis court used for singles play than by the junior court? c. The court for players 10 and under has the same width but is 18 feet shorter than the court for singles play. How much more area is covered by the tennis court used for singles play? 14. Draw Conclusions The cross section of a metal ingot is a trapezoid. The cross section has an area of 39 square centimeters. The top base of the cross section is 1 centimeters. The length of the bottom base is centimeters greater than the top base. How tall is the metal ingot? Explain. FOCUS ON HIGHER ORDER THINKING Work Area 15. Analyze Relationships A mirror is made of two congruent parallelograms as shown in the diagram. The parallelograms have a combined area of 9 1_ square yards. The height of each parallelogram is 1 yards. 3 1_ 3 a. How long is the base of each parallelogram? b. What is the area of the smallest rectangle of wall that the mirror could fit on? 16. Persevere in Problem Solving A watercolor painting is 0 inches long by 9 inches wide. Ramon makes a mat that adds a 1-inch-wide border around the painting. What is the area of the mat? 9 in. 1 in. 0 in. 1 yd 464 Unit 5

23 ? LESSON 16.4 ESSENTIAL QUESTION Solving Volume Equations Problem Solving by Finding Volume To find the volume of a box, which is in the shape of a rectangular prism, you can multiply the length, the width, and the height. The volume of a threedimensional shape is always in cubic units, such as cubic meters or cubic inches. Volume of a Rectangular Prism Equations, expressions, and relationships 6.8.C Write equations that represent problems related to... volume of right rectangular prisms where dimensions are positive rational numbers. Also 6.8.D How do you write equations to solve problems involving volume of right rectangular prisms? Math On the Spot The volume V of a rectangular prism is the product of its length l, its width w, and its height h. V = lwh l w h Math Talk Mathematical Processes Why are V = lwh and V = Bh both formulas for the volume of a rectangular prism? EXAMPLE D A rectangular swimming pool is 5 meters long and 17 1_ meters wide. It has an average depth of 1 1_ meters. What is the volume of the pool? Label the rectangular prism to represent the pool. l = 5 meters w = 17 1_ meters h = 1 1_ meters Use the formula to write an equation. V = lwh V = _ 1 1_ 5 m m m = 5 35 _ 3 =,65 4 = 656 1_ cubic meters 4 Write mixed numbers as fractions greater than 1. Multiply. Write as a mixed number in simplest form. Lesson

24 YOUR TURN Online Assessment and Intervention Math On the Spot 1. Miguel has a toolbox that measures 18 1_ inches by 1 1_ inches by 4 inches. What is the volume of the toolbox? V = cubic inches Writing Equations Using the Volume of a Rectangular Prism You can use the formula for the volume of a rectangular prism to write an equation. Then solve the equation to find missing measurements for a prism. EXAMPLE 6.8.C Samuel has an ant farm with a volume of 375 cubic inches. The width of the ant farm is.5 inches and the length is 15 inches. What is the height of Samuel s ant farm? V = lwh 375 = 15.5 h 375 = 37.5h Write the formula. Use the formula to write an equation. Multiply. Online Assessment and Intervention = 37.5h = h The height of the ant farm is 10 inches. Reflect. Communicate Mathematical Ideas Explain how you would find the solution to Example using the formula V = Bh. YOUR TURN Divide both sides of the equation by Find the height of this shape, which has a volume of cubic feet. 3 4 ft? 1 ft Image Credits: Thom Lang/CORBIS 466 Unit 5

25 Solving Multistep Problems One cubic foot of water equals approximately 7.5 gallons and weighs approximately 6.43 pounds. EXAMPLE D Math On the Spot The classroom aquarium holds 30 gallons of water. It is 0.8 feet wide and has a height of feet. Find the length of the aquarium. STEP 1 Find the volume of the classroom aquarium in cubic feet. 30 gallons = 4 cubic feet 7.5 gallons per cubic foot The volume of the classroom aquarium is 4 cubic feet. Divide the total number of gallons by the unit rate to find the number of cubic feet. STEP Find the length of the aquarium. V = lwh 4 = l = l(1.6) = l(1.6) = l Write the formula for volume. Use the formula to write an equation. Multiply. Divide both sides of the equation by 1.6. Reflect The length of the classroom aquarium is.5 feet. 4. Persevere in Problem Solving How much does the water in the classroom aquarium weigh? Explain. YOUR TURN 5. An aquarium holds gallons of water. It has a length of feet and a height of 1.5 feet. What is the volume of the aquarium? What is the width of the aquarium? Explain. Online Assessment and Intervention Lesson

26 Guided Practice 1. Find the volume of this rectangular prism. (Example 1) V = lwh 1 1 in. V = 3 1_ 3 1 in. V = V = in. The volume of the rectangular prism is cubic inches.. Write an equation to find the width of the rectangular prism. Show your work. (Example ) V = 6,336 cm 3 18 cm 16 cm? cm 3. One red clay brick weighs 5.76 pounds. The brick is 8 inches long and 1_ 4 inches wide. If the clay weighs 0.08 pounds per cubic inch, what is the volume of the brick? Write an equation to find the height of the brick. Show your work. (Example 3)? ESSENTIAL QUESTION CHECK-IN 4. How do you solve problems about volume of right rectangular prisms? 468 Unit 5

27 Name Class Date 16.4 Independent Practice 6.8.C, 6.8.D 5. Jala has an aquarium in the shape of a rectangular prism with the dimensions shown. What is the height of the aquarium? Online Assessment and Intervention? Height = 6. Find the volume of a juice box that is 3 in. by 1 1_ in. by 4 in. Volume = 7. Find the width of a cereal box that has a volume of 3,600 cm 3 and is 0 cm long and 30 cm high. Width = V = 3, cubic inches 4.5 in. 1.5 in. 8. Bill has a box of markers that has a base of 8 cm by 0 cm and a height of 6 cm. Martin s pencil box has a height of 4 cm and a base that is 15 cm by 16 cm. Bill says his marker box has the same volume as Martin s pencil box. Is Bill right? Explain. 9. Physical Science A small bar of gold measures 40 mm by 5 mm by mm. One cubic millimeter of gold weighs about ounces. Find the volume in cubic millimeters and the weight in ounces of this small bar of gold. 10. History The average stone on the lowest level of the Great Pyramid in Egypt was a rectangular prism 5 feet long by 5 feet high by 6 feet deep and weighed 15 tons. What was the volume of the average stone? How much did one cubic foot of this stone weigh? 11. A freshwater fish is healthiest when there is at least one gallon of water for every inch of its body length. Roshel wants to put a goldfish that is about 1_ inches long in her tank. Roshel s tank is 7 inches long, 5 inches wide, and 7 inches high. The volume of 1 gallon of water is about 31 cubic inches. a. How many gallons of water would Roshel need for the fish? b. What is the volume of Roshel s tank? c. Is her fish tank large enough for the fish? Explain. Lesson

28 1. A box of crackers is a rectangular box with the dimensions shown. The box is one-fourth full. What is the volume of crackers in the box? FOCUS ON HIGHER ORDER THINKING 8 in. 4 in. 4 in. Work Area 13. Multistep Larry has a clay brick that is 7 inches long, 3.5 inches wide, and 1.75 inches thick, the same size as the gold stored in Ft. Knox in the form of gold bars. Find the volume of this brick. If the weight of the clay in the brick is 0.1 pound per cubic inch and the weight of the gold is 0.7 pounds per cubic inch, find the weight of the brick and the gold bar. Round all answers the nearest tenth. Volume of the brick or bar = cubic inches Weight of the brick = Weight of the gold bar = pounds pounds 14. Represent Real-World Problems Luisa s toaster oven, which is in the shape of a rectangular prism, has a base that is 55 cm long by 40 cm wide. It is 30 cm high. Luisa wants to buy a different oven with the same volume but a smaller length, so it will fit better on her kitchen counter. What is a possible set of dimensions for this different oven? 15. Multiple Representations Use the formula V = Bh to write a different version of this formula that you could use to find the area of the base B of a rectangular prism if you know the height h and the volume V. Explain what you did to find this equation. 16. Communicate Mathematical Ideas Explain how you could find the volume of a cube that has an edge of e. 17. Justify Reasoning Mariel says that a jewelry box that is 3 inches high, 4 1_ inches wide, and 5 inches long has a volume of 67 1_ inches. Katy says that answer is not quite correct. What is the error in Mariel s answer? 470 Unit 5

29 MODULE QUIZ Ready 16.1 Area of Quadrilaterals 1. Find the area of the figure. 1 1 yd Online Assessment and Intervention 16. Area of Triangles 1 17 yd 5. Find the area of the triangle. 14 ft 16.3 Solving Area Equations 17 ft 3. A triangular pane of glass has a height of 30 inches and an area of 70 square inches. What is the length of the base of the pane? 4. A tabletop in the shape of a trapezoid has an area of 6,550 square centimeters. Its longer base measures 115 centimeters, and the shorter base is 85 centimeters. What is the height? 16.4 Solving Volume Equations 5. A rectangular shoebox has a volume of 78 cubic inches. The base of the shoebox measures 8 inches by 6.5 inches. How long is the shoebox? ESSENTIAL QUESTION 6. How can you use equations to solve problems involving area and volume? Module

30 MODULE 16 MIXED REVIEW Texas Test Prep Online Assessment and Intervention Selected Response 1. What is the area of the rhombus shown below? 3 in. 4. The trapezoid below has an area of 1,575 cm. 63 cm 8 in. 7 cm Which equation could you solve to find the height of the trapezoid? A 45h = 1,575 C 850.5h = 1,575 B 90h = 1,575 D 1,701h = 1,575 A 161 in C 644 in B 3 in D 966 in. What is the area of the triangle shown below? Gridded Response 5. Cindy is designing a rectangular fountain in a courtyard. The rest of the courtyard will be covered in stone. ft 3.7 mm 1 ft 6 ft Courtyard Fountain 4.8 mm A 4.44 mm B 5.9 mm C 8.88 mm D mm 3. A rectangular prism has a volume of 91 cubic meters. It has a length of 19 meters and a width of 1 meters. Which equation could be solved to find the height of the rectangular prism? A 114h = 91 B 8h = 91 C 15.5h = 91 D 31h = 91 The part of the courtyard that will be covered in stone has an area of 46 ft. What is the width of the fountain in feet? Unit 5