An hourglass, composed of two identical cones, is 12 cm tall. The radius of each cone is 3 cm. If you want to fill the bottom half of the hourglass full of salt, how much salt will you need? Explain the method you would use to find the amount of salt.
bottom half is a cone. r=3, h=6
v = 1/2 pi r^2 h
Now just plug in the numbers
oops. 1/3 pi r^2 h
37.68 cm^3
To find the amount of salt needed to fill the bottom half of the hourglass, we need to calculate the volume of that section. Here's how you can do it:
1. Visualize the hourglass: Start by visualizing the hourglass in your mind. Remember that it is formed by two identical cones attached together at their bases.
2. Identify the relevant measurements: Take note of the height of the hourglass (12 cm) and the radius of each cone (3 cm).
3. Calculate the volume of each cone: The volume of a cone is given by the formula V = (1/3) * π * r^2 * h, where V is the volume, π is Pi (approximately 3.14159), r is the radius, and h is the height.
For each cone, the radius (r) is 3 cm, and the height (h) is half of the total height of the hourglass (12 cm / 2 = 6 cm). Plug these values into the formula to find the volume of one cone:
V1 = (1/3) * π * (3 cm)^2 * (6 cm)
4. Calculate the volume of the bottom half: Since the hourglass is formed by two identical cones, the volume of the bottom half will be the sum of the volumes of each cone.
Volume of the bottom half = V1 + V1
5. Simplify and calculate: Add the volume of one cone to itself to find the volume of the bottom half of the hourglass.
Volume of the bottom half = 2 * V1
6. Substitute values and calculate: Substitute the known values into the formula and use a calculator to find the result.
Volume of the bottom half = 2 * (1/3) * π * (3 cm)^2 * (6 cm)
By evaluating this expression, you should obtain the volume of the bottom half of the hourglass. To find the amount of salt needed, you can multiply this volume by the density of the salt you plan to use. Keep in mind that the final result will depend on the units used for density (e.g., grams per cubic centimeter) and the desired quantity of salt (e.g., grams or kilograms).