Exercises

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.