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 2/3 full of salt, how much salt will you need? Explain the method you would use to find the amount of salt.

How would I solve this?

Actually it would be 37.7 cm^3 because the height of an individual cone is 6

1/3 (pi *r^2)(h) ?

So you want 2/3 of one cone

volume = (2/3)(1/3)π(3^2)(12)
= 24π cm^3

75.40 cm^3 would that be the final answer?

Well, let's break it down and sand-ify the situation! The bottom half of the hourglass, my friend, is essentially just one of those identical cones, right? Now, we need to figure out how much salt we'll need for that.

To calculate the volume of a cone, you use the formula: V = (1/3)πr²h, where V is the volume, r is the radius, and h is the height. So, in this case, the height of our cone (which is the same as the hourglass) is 12 cm, and the radius is 3 cm.

Let's plug those values into our handy-dandy formula, shall we? V = (1/3)π(3²)(12). Now, simplify that equation, and we get V = 36π cm³.

But hold your clown makeup, we're not done yet! We only want to fill the bottom half of the hourglass 2/3 full of salt. That means we just need 2/3 of the calculated volume.

To get this, we multiply our volume by the fraction 2/3: (2/3)(36π) cm³.

And there you have it! That's the amount of salt you'll need to fill the bottom half of the hourglass. Just don't forget to take some breaks and make funny sandcastle sculptures along the way!

To solve this problem, we can break it down into smaller steps:

Step 1: Determine the volume of the entire hourglass:
Since the hourglass is composed of two identical cones, we need to find the volume of one cone and then double it. The formula for the volume of a cone is given by V = (1/3) * π * r^2 * h, where r is the radius of the cone and h is the height.

Plugging in the values, we have:
V_cone = (1/3) * π * (3 cm)^2 * 12 cm
V_cone = (1/3) * π * 9 cm^2 * 12 cm
V_cone = (1/3) * π * 108 cm^3

Since we have two cones, the volume of the entire hourglass is:
V_hourglass = 2 * V_cone
V_hourglass = 2 * (1/3) * π * 108 cm^3
V_hourglass = (2/3) * π * 108 cm^3

Step 2: Determine the volume of the bottom half of the hourglass:
We want to fill the bottom half of the hourglass 2/3 full of salt. This means we need to find 2/3 of the volume of the bottom half of the hourglass.

V_bottom_half = (2/3) * V_hourglass
V_bottom_half = (2/3) * (2/3) * π * 108 cm^3
V_bottom_half = (4/9) * π * 108 cm^3

Step 3: Find the amount of salt needed:
To find the amount of salt needed, we assume that the salt fills the entire volume without any gaps. So the volume of salt needed is equal to the volume of the bottom half of the hourglass.

Amount of salt needed = V_bottom_half
Amount of salt needed = (4/9) * π * 108 cm^3

To calculate the final amount of salt needed, you can substitute the value of π (pi) as 3.14 in the formula and perform the calculation.

Amount of salt needed ≈ (4/9) * 3.14 * 108 cm^3

By simplifying this expression, you can find the amount of salt needed in cubic centimeters.