Exercises

Lesson 1: The Pythagorean Theorem

1. Leila is building a wooden wheelchair ramp for her grandma's house. The ramp must rise \(1.5\) m over a horizontal distance of \(6\) m. How long will the wooden surface of the ramp be?

2. Mackenzie wants to buy a round table for her new apartment. She measures the door to her apartment to be \(2\) m high and \(75\) cm wide. What is the diameter of the largest round table that Mackenzie can buy so that it will fit through her doorway?

3. Sarah leaves her home and drives south \(150\) km. Then she drives west \(70\) km. At this point, she drives north \(30\) km. Finally, she drives east \(20\) km. Draw a diagram of Sarah's trip. At the end of the trip, what is the straight line distance from Sarah's car to her home?

4. Determine the value of \(x\) in the diagram.

   

5. A triangle has side lengths \(4\), \(5\), and \(6\). 

   1. Do these side lengths form an acute triangle or an obtuse triangle?

   2. Determine which one of the lengths to change (and what to change it to) in order to create a right triangle with integer side lengths.

6. A warehouse is installing a pulley system from the corner of the floor, \(X\), to the opposite corner on the ceiling, \(Y\). This will require a rope that runs directly from \(X\) to \(Y\) and directly back to \(X\) in a straight line.

   The warehouse is \(100\) m by \(50\) m by \(20\) m. 

   How much rope is needed for the pulley system? 

   

7. The area of a right isosceles triangle is \(100\) cm². What is its perimeter? 

8. Triangle \(ABC\) has vertices \(A\left(-1,5\right)\), \(B\left(3,2\right)\), and \(C\left(-3,-3\right)\). Graph the triangle then determine its perimeter.

9. An electronics company is marketing their new "Super-Duper Widescreen" technology, which has an aspect ratio of \(12:5\).

   1. What would be the dimensions of a \(65\)-inch "Super-Duper Widescreen" monitor?

   2. Do you think that it is a good idea to use only the diagonal length to represent the size of a screen? Justify your response.

Lesson 2: Perimeter and Area of Composite Shapes

Questions 1 and 2 refer to the following figure.

1. Calculate the perimeter of the figure in two different ways: one that involves determining the lengths of all the segments in the diagram, and one that does not.
2. Calculate the area of the figure in two different ways: one that involves addition and one that involves subtraction.

Questions 3 to 5 refer to the following information.

Michael is building a swimming pool in the shape of a rectangle with a semi-circle attached to the shorter side. The dimensions of the rectangular part of the pool is \(4.3\) m by \(8.5\) m. The pool will be surrounded on all sides by a cement deck \(1\) m wide.

3. Michael wants to buy a pool cover that will just cover his pool.
   If the pool cover material costs \($11.50\)/m², what will be the total cost of the cover?
4. What is the area of the cement needed to create the deck?
5. Michael needs to build a fence around the entire pool and deck area. He already has a \(1.5\) m wide gate but needs to buy the rest of the fencing. What length of fencing does he need?
6. The shape below has an area of \(100\) cm². What is the perimeter of the shape?

   

7. In Figure 1, a circle with radius \(10\) is inscribed in a square. In Figure 2, a square is inscribed in a circle of radius \(10\). In each figure, the area between the two shapes is shaded. Which figure has the greater shaded area?

   Figure 1

   

   Figure 2

   

8. Hunter has two shapes, Shape 1 and Shape 2. He can makes a Composite shape by adding Shape 2 to Shape 1 or by subtracting Shape 2 from Shape 1.
   1. What could Shape 1 and Shape 2 be so that the perimeter of the Composite shape is less than the perimeter of Shape 1?
   2. What could Shape 1 and Shape 2 be so that the perimeter of the Composite shape is equal to the perimeter of Shape 1?
   3. What could Shape 1 and Shape 2 be so that the perimeter of the Composite shape is greater than the perimeter of Shape 1?

Questions 9 and 10 refer to the following information.

A composite shape is formed by two overlapping congruent circles with radius \(8\) cm and centres \(A\) and \(C\). The circles intersect at points \(B\) and \(D\) so that \(ABCD\) is a square.

9. Determine the perimeter of the composite figure.
10. Calculate the area of the composite figure.

Lesson 3: Surface Area of Pyramids and Cones

1. Hexagonal Pyramid A has slant height \(30.1\) m, base edge \(3.0\) m, and apothem \(2.6\) m.

   Hexagonal Pyramid B has slant height \(7.0\) m, base edge \(8.0\) m, and apothem \(6.9\) m.

   Which pyramid has the greater surface area and by how much? Round your answer to \(1\) decimal place.

2. What is the surface area of a right pyramid with a rectangular base of \(6\) cm by \(8\) cm and a height of \(5\) cm?

3. Paige is throwing a birthday party and wants to make cone shaped party hats. She wants the radius of the hat to be \(15\) cm and the height to be \(36\) cm. How much paper does she need to make \(4\) hats?

   

   Source: Party Hats - mashabuba/E+/Getty images

4. Christal is making a conical paper popcorn container. She would like the cone to have a height of \(25\) cm and its open edge to have a circumference of \(55\) cm. She is going to trace out a large circle on a piece of paper, cut out the circle as well as cut along the radius to the centre, and then roll up the piece of paper to make the cone.
   1. What will the radius of the large circle have to be in order to create the container?
   2. What will be the area of the paper used to make the container?
5. Is it possible for the base area of a pyramid to equal its lateral area? Explain why or why not.
6. The surface area of a square-based pyramid is \(1804\) cm² and its base side length is \(22\) cm.
   1. Calculate its slant height.
   2. Calculate its vertical height.
7. An octagonal-based pyramid has base side lengths of \(20\) cm and slant height of \(80\) cm. What is the surface area of the pyramid?

8. Heather enjoys creating stained glass art. Her next project is to make a lampshade based on the frustum of a square-based pyramid. The side length of the base of the pyramid is \(42\) cm, the height of the entire pyramid would be \(28\) cm, and the actual height of the frustum shade is \(24\) cm.

   Determine the total area of the glass used for the lampshade.

9. A regular tetrahedron is a pyramid that has congruent equilateral triangles as its base and lateral surfaces.
   1. In order to calculate the surface area of a regular tetrahedron, we don't need as many measurements as we would for other types of regular pyramids. Why is this?
   2. Two pieces of information: A regular tetrahedron has edge length \(18\) cm and slant height \(15.6\) cm. Calculate its surface area.
   3. One piece of information: A regular tetrahedron has edge length \(12\) cm. Calculate its slant height and its surface area.

Lesson 4: Volume of Pyramids and Cones

1. Kiah wants to make \(12\) hexagon-based pyramid candles. The mold she has will make candles with a base side length of \(2\) cm, an apothem length of \(1.7\) cm and a height of \(18\) cm. She can buy a block of beeswax that has a volume of \(230\) mL for \($11\). How much will it cost Kiah to make the \(12\) candles?
2. A sugar cone has a volume of \(70.5\) cm³ and a diameter of \(4.8\) cm. What is the height of the sugar cone?
3. A glass timer is made of two cones placed tip to tip on top of each other. The sand in the timer funnels down at a rate of \(45\) mm³/s. As the sand begins to fall, the depth of the sand is \(3\) cm and the diameter of the top of the sand is \(1.6\) cm. How much time does the timer measure?

4. In July 2018, a teepee was set up at the Healing Camp for Justice in Saskatoon, Saskatchewan, Canada. The thirteen poles of the frame were covered with canvas to make the teepee. Suppose the shape of the teepee was a regular right pyramid with an apothem of \(5\) feet, base edges of \(2.46\) feet, and slant height of \(15\) feet, calculate its volume.

   

5. A cylinder and a cone have the same volume and the same radius. What is the height of the cone?
6. A dump truck is carrying \(15~000\) kg of gravel, which has a density of about \(1600\) kg/m³. The truck delivered some of the gravel at a work site, leaving a conical pile with a base circumference of \(12\) m and a height of \(1.5\) m. What is the mass of the gravel that remained in the truck?
7. The shape of a crayon is a cylinder with a cone tip. The height of the crayon is \(9\) cm with a diameter of \(0.8\) cm. The tip has a height of \(1.3\) cm. A \(24\)-pack of crayons is packaged in a rectangular box with dimensions \(2.5 \text{ cm} \times 6.5 \text{ cm} \times 9.2 \text{ cm}\). How much empty space is in a full \(24\)-pack box?

8. A regular right octahedron is a geometric solid with eight congruent equilateral triangular faces. Determine the volume of a regular octahedron whose edges have a length of \(80\) cm.

   

9. In chemistry, a common container used is the Erlenmeyer flask. The shape of an Erlenmeyer flask is approximately a cone with its top removed for the bottom portion of the flask and a cylinder for the neck.

   An Erlenmeyer flask has a base diameter of \(8.2\) cm, a total height of \(12.8\) cm, a neck diameter of \(3.6\) cm and a neck height of \(3.6\) cm. The height of the entire cone would have been \(16.4\) cm. What is the capacity of the flask?

   

10. A block toy is made of three pieces that form a pentagonal pyramid when stacked on top of one another. The toy's base has edge length \(10.8\) cm and apothem \(7.5\) cm. Its height is split into three layers each \(6\) cm high. Calculate the volume of each of the three pieces of the toy.

    

Lesson 5: Volume and Surface Area of Spheres

Round all answers to 1 decimal place.

1. Kalam has a mold for making spherical popsicles, each with a diameter of \(4.5\) cm. She has \(1\) L of fruit juice. Does Kalam have enough juice to make \(24\) popsicles?
2. Snow cones are served at a summer carnival in conical cups with a radius of \(5\) cm and a height of \(10\) cm. The cup is filled with snow cone slush then an extra hemisphere of slush is piled on top. If the hemisphere and the cup have the same radius, calculate the volume of slush in one snow cone.
3. The Dominion Astrophysical Observatory in Victoria, British Columbia, is a cylindrical shape with a dome shaped roof. The diameter of the cylinder is \(20.2\) m and the height is \(22.3\) m. The outside of the structure needs to be repainted. What is the surface area that needs to be painted, including the roof?

   

4. A hemisphere is half of a sphere.
   1. Calculate the surface area of a hemisphere with diameter \(11\) cm.
   2. Show how to develop a general formula for the surface area of a hemisphere in terms of its radius.

   

5. Jamal is making an aromatic ornament by pushing cloves into the peel of a spherical orange with a diameter of \(8\) cm. The part of each clove that will be visible on the surface of the orange is approximately a square with side lengths of \(5\) mm.

   How many cloves will Jamal need if he wants to completely cover the surface of the orange?

   

6. A manufacturer is deciding whether to use a rectangular prism or a cylinder to package three tennis balls with a radius of \(3.3\) cm. Which package would have less empty space?
7. 1. A sphere and a cone have the same volume and radius. What is the height of the cone \(h\) in terms of the radius \(r\)?
   2. A large spherical balloon has three times the volume of a smaller spherical balloon. What is the radius of the large balloon \(R\) in terms of the radius of the small balloon \(r\)?

8. Liam is creating a three dimensional model of a head. He uses a \(7.4\) L styrofoam ball and removes one-sixth of the ball by cutting a wedge whose tip is a diameter of the sphere. What is the surface area that needs to be painted on his model? Include the inside of the “mouth”.

   

9. The Earth is made up of an inner core, outer core, mantle, and crust. The crust makes up about \(1\%\) of the Earth’s volume. The radius of the Earth is \(6371\) km. What is the depth of the Earth's crust?

   

   Source: Earth's Layers - bubaone/iStock/Getty Images

10. Planet Aqua is completely covered by an ocean of a consistent depth. The radius to the ocean floor of the planet is \(2000\) km. The volume of the water in the ocean is equal to the volume of material in the planet itself. Calculate the depth of the ocean.

Lesson 6: Maximizing Area of Rectangles With Fixed Perimeter

1. The perimeter of a piece of paper is \(18\) cm.
   1. List three different possible sets of dimensions for the paper, as well as the resulting areas.
   2. What are the dimensions that result in the maximum possible area? What is the maximum area?

2. A playground has a tetherball pole. The tetherball court requires a circular area of \(19.6\) m². If the camp uses \(20\) m of fencing to create a rectangular enclosure with the maximum area, do they have enough fencing to enclose the tetherball court? 

   

3. Sandeep has roped off a square area of \(18\) m² in his backyard. He wants to move roping to create a rectangle using his house as one of the sides. What dimensions will the new rectangle have if he wants to maximize the area? Round your answers to two decimal places.
4. A rectangular bike parking enclosure is delineated by \(32\) m of rope on three sides and a wall on the fourth. What is the maximum possible area of the enclosure?
5. Two fenced enclosures are being constructed against a wall, each using \(12\) m of fencing. One enclosure is a semicircle and the other is a rectangle with maximized area. Which enclosure has the greater area and by how much? Round your answer to one decimal place.
6. Vilma currently keeps her chickens in a fenced circular pen with area \(100\) m². If she re-shapes the pen into a rectangle, what would be its maximum possible area? Round your answer to one decimal place.
7. A dog kennel company has a rectangular yard that will be partitioned into \(3\) congruent pens using \(2\) partitions parallel to the length of the yard. The company has \(200\) m of fencing for the exterior of the pens and the partitions. What dimensions for the yard will maximize the area?
8. Alireza is designing some riverside pens for his pets. He would like his rabbits and his guinea pigs to have congruent pens, with the river on one side and fencing on the other sides. The pens will be side-by-side so that they share a side of fencing.

   

   Alireza has \(60\) metres of fencing to use. What is the maximum possible area for the pens and what are the dimensions of the pens that produce the maximum area?

9. Goldenrose is building a greenhouse in the shape of a square-based prism. The frame will be made with heavy duty plastic pipe. She has \(60\) m of pipe that will be cut into the eight edges necessary to build the greenhouse.

   

   Goldenrose does some research and discovers that the volume of such a structure is maximized when each horizontal piece is one-sixth the total length of pipe available.

   1. What are the dimensions of the greenhouse that result in the maximum volume?
   2. What is the maximum volume of the greenhouse? 

Lesson 7: Determining the Optimal Perimeter of Rectangles

1. Bruslan is the architect in charge of designing a new elementary school. He has been told that each rectangular classroom should include \(25\) m² of common space, plus \(1.50\) m² per student for desks and chairs. If the maximum number of students in a class is \(22\), what is the least possible perimeter that Bruslan's classroom design could have? Round your answer to one decimal place.
2. Jasneet has a \(4\) m by \(5\) m rectangular garden plot in her backyard. She wants to double the area of her garden this year but minimize the amount of fencing she needs to enclose the garden on all sides. By how much should she increase the dimensions of her new garden? Round your answers to one decimal place.
3. The perimeter is minimized for a rectangle with an area of \(35\) cm². What is the area of a circle that has the same perimeter as the rectangle? Round your answer to one decimal place.
4. Lovisa has \(1512\) tiles, each measuring \(1\) inch by \(1\) inch. She is going to use the tiles to cover a rectangular tabletop then put a decorative border along the outer edge. Assume that Lovisa will use all of the tiles and that they cannot be cut into smaller pieces. Determine the dimensions of the tabletop that will minimize the length of the decorative border.
5. A community beach wants to rope off a swimming area of \(400\) m². Should they create a semicircular or rectangular area if they want to minimize the roping needed?

   

6. Tjah has a rectangular field for his horses with a fence on three sides and a river as the fourth side. The present dimensions of the field are \(30\) m by \(30\) m, but the fence is old and needs replacing. Tjah thinks that changing the dimensions of the field will save him time and fencing. How much less fencing will Tjah need if the same area is enclosed in the most efficient way? Round your answer to one decimal place.
7. Varsha is planning a \(20\) m² rectangular pen for her rabbits. The fencing she likes costs \($3.49\) per metre, and must be purchased in half-metres. She is considering a free-standing (enclosed on four sides) pen compared to one that is built using a wall as one of the sides. What would be the difference in cost between these two options, assuming each one is planned in the most optimal way?

8. Two adjacent, congruent, partially-enclosed pens have a river on the fourth side and a combined area of \(100\) m². Use a table of values or a spreadsheet to determine the minimum length of fencing, to one decimal place, needed to construct the pens.

   

9. Pharell is planning to build a \(128\) m² deck against his house. He wants to enclose the deck using glass panels along the outer edges of the deck. To minimize his costs, he wants to use the least amount of glass panelling. 

   1. If the deck is rectangular with the house on one side, what length of the deck’s edge will need glass panelling?

      

   2. Pharell considers making an L-shaped deck on the corner of the house. Two congruent short  edges would be touching and perpendicular to the house walls while two congruent longer edges would be parallel to the house walls. What dimensions should he use to minimize the length of the deck’s edge that will need glass panelling and what is the minimum length of the deck's edge? Round your answers to one decimal place.

      

   3. Which configuration should Pharell choose to minimize the length of the deck’s edge?

Lesson 8: Optimizing Surface Area of Cylinders and Square-Based Prisms

1. Wenqian is planning to buy some angelfish and needs to design the size and shape of the square-based aquarium they will live in. His research shows that there should be about \(8\) L of water per fish. In order for the water to lose as little heat as possible, the surface area of the aquarium should be minimized. What are the dimensions of the square-based aquarium that will support \(5\) fish and minimize heat loss? Round your answers to one decimal place.

   

2. Selina wants to design a travel mug that will keep her coffee hot for the longest amount of time. She wants it to be cylindrical and hold a volume of \(591\) mL. What dimensions should Selina use for the travel mug? Round your answers to one decimal place.

   

3. Forty-eight QuadCube puzzles are shipped in a square-based box. The dimensions for one QuadCube are \(5.7\) cm by \(5.7\) cm by \(5.7\) cm. What are the most efficient dimensions of the box?
4. Ice has a density of \(0.92\) g/cm³. What dimensions would minimize the melting of a \(1\) kg block of ice? Round your answers to one decimal place.
5. A can of soda contains \(355\) mL of liquid. 
   1. What dimensions would minimize the amount of aluminum needed to make the cans? Round your answers to one decimal place.
   2. A manufacturer currently makes cans with a diameter of \(5.8\) cm and a height of \(13.4\) cm. If the manufacturer switches to making cans with the dimensions found in part a), what percent of aluminum would be saved?
6. Beans come in cylindrical \(300\) mL cans and these are packed in square-based boxes with \(36\) cans each. Assuming the most efficient packaging for both the cans and the cases, what are the dimensions of the box? Round your answers to one decimal place.
7. A company that produces soup has hired you as a consultant to help them decide whether to package their soup in cylindrical or square-based prism containers.
   1. Each container is to hold \(1\) L of soup and the packaging material costs \($0.03\) per \(10\) cm². What is the difference in cost between the optimal cylindrical container and the optimal square-based prism container? Round your answer to the nearest cent.
   2. The company needs to ship the soup containers to stores in trucks. Explain which container is more efficient to pack and transport.
8. A movie theatre is designing an open-top container for their medium popcorn. They want the container to have a capacity of \(3.5\) L. 

   1. What dimensions for a square-based prism should they use to minimize the amount of packaging needed? Round your answers to one decimal place.

      

   2. What dimensions for a cylinder should they use to minimize the amount of packaging needed? Round your answers to one decimal place.

      

9. Examine the answers in the previous question.
   1. Make a conjecture for a formula that could be used to determine the optimal dimensions of an open-top square-based prism given its volume \(V\).
   2. Make a conjecture for a formula that could be used to determine the optimal dimensions of an open-top cylinder given its volume \(V\).

Lesson 9: Maximizing the Volume of Cylinders and Square-Based Prisms

1. Pearl is making square-based prism gift boxes for a party, each using \(900\) cm² of cardboard. What dimensions should she use for the boxes if she wants to maximize how much the boxes will hold? Round your answers to 1 decimal place.
2. A cylinder has the maximum volume for its surface area of \(480\) cm². What is the height of the cylinder? Round your answer to 1 decimal place.
3. Rainbow Bright Paint Company has a \(3000\) L vat of paint to package in cylindrical cans. Each can has a surface area of \(2200\) cm². What is the minimum number of paint cans needed to contain all of the paint in the vat?
4. Jiordano makes gourmet jellybeans packaged in square-based prism boxes. The box dimensions are \(7\) cm by \(7\) cm by \(12\) cm. Jiordano's friends suggest that the boxes could be shaped to have a greater capacity.
   1. What is the greatest increase in volume that could be achieved using the same amount of packaging material to create a square-based box with different dimensions? Round your answer to 1 decimal place.
   2. Describe two factors that Jiordano may consider for keeping the original box dimensions.
5. Given the same surface area, which holds more, a square-based prism or a cylinder?
6. What is the maximum number of sugar cubes that can be contained in a square-based box of a surface area \(700\) cm² if each sugar cube has side lengths of \(1.5\) cm?
7. Milaan is designing a dessert made of a chocolate cylinder shell with a mousse filling. She has enough chocolate to cover an area of \(1200\) cm² and she wants the chocolate cylinders to hold as much mousse as possible.
   1. Determine the dimensions of the cylinders if Milaan makes \(6\) desserts. Round your answers to one decimal place.
   2. Calculate the volume of the mousse she would need, to the nearest millilitre.

8. Dax wants to build a terrarium using \(5400\) cm² of glass. The shape of the terrarium will be a square-based prism without a lid but with the maximum volume. What are the dimensions and volume of the terrarium?

   

9. A square-based box made of card stock requires \(1\) cm of overlap on one of every pair of raw edges, as shown in the net. The total area of the material used, including the overlaps, is \(600\) cm².

   1. Use a table of values or a spreadsheet to determine the dimensions of the box that will result in the maximum volume. Round your answers to 1 decimal place.
   2. What is the maximum volume? Round your answer to 1 decimal place.

   

10. You are given a standard letter size piece of paper, \(8.5\) inches by \(11\) inches. You are only allowed to cut two circles and a rectangle to make the cylinder. If the cylinder is to have the maximum volume possible, determine its dimensions. Round your answers to 2 decimal places.