Water is being poured into a conical cup that is 12cm tall.
a) When the water in the cup is 9cm deep, what percentage of the cup is filled?
b) When the cup is 75 percent filled, how deep is the water?
But what about the radius
The cup is a cone, not a cylinder.
9/12 = 3/4, so the volume is (3/4)^2 = 9/16 = 56.25%
When 75% is full, the depth is √.75 = 0.866, or 10.39 inches
a) Hahaha, well, let's do some math here. Since the cup is conical, we can calculate its volume using the formula V = (1/3)πr²h, where r is the radius and h is the height. Given that the cup is 12cm tall, and the water is 9cm deep, we can find the radius of the water level by using similar triangles. Since the heights are in a 3:4 ratio (9cm to 12cm), the radii will also be in a 3:4 ratio. Therefore, the radius of the water level is (3/4)*r, and we can calculate the volume of the water using the formula. To find the percentage of the cup filled, we just need to divide the volume of the water by the volume of the entire cup and multiply by 100. Okay, okay, let me grab my calculator! crunch, crunch, beep, boop... Oh! Looks like the cup is filled approximately 56.25%! I hope that makes you smile!
b) Ah, 75 percent... well, let's put those clown math skills to work again! To find how deep the water is when the cup is 75 percent filled, we need to determine the height. Since we know the cup is 12cm tall, we can multiply 12 by 75% (or 0.75) to find the depth of the water. Let's do some circus calculations! Ta-da! The water will be approximately 9cm deep when the cup is 75 percent filled. Keep filling it up, but don't spill any water on your clown shoes!
To solve the given problem, we need to understand the concept of volume and how it relates to the shape of a cone.
The volume of a cone can be calculated using the formula V = (1/3)πr²h, where V represents the volume, r represents the radius of the base, and h represents the height.
Now, let's break down the problem into two parts, starting with the first question:
a) When the water in the cup is 9cm deep, what percentage of the cup is filled?
To find the percentage of the cup filled, we need to compare the depth of the water to the total height of the cup.
Given information:
- Height of the cup (h) = 12 cm
- Depth of the water (d) = 9 cm
To find the volume of water in the cup, we can subtract the remaining empty part from the total volume of the cone.
Empty part (height) = h - d = 12 cm - 9 cm = 3 cm
To calculate the filled volume:
V_filled = (1/3)πr²d
To calculate the total volume:
V_total = (1/3)πr²h
To find the percentage of the cup filled, we can use the formula:
Percentage filled = (V_filled / V_total) * 100
b) When the cup is 75 percent filled, how deep is the water?
We need to find the depth of the water when the cup is 75 percent filled. Again, we have the total height of the cup (h) and we need to find the depth (d).
Given information:
- Height of the cup (h) = 12 cm
- Percentage filled = 75%
To find the depth of the water:
d = (Percentage filled / 100) * h
Now that we have the formulas and necessary calculations, we can substitute the values and solve for the desired quantities.