Let's tackle each part of your questions step by step, starting with informal reasoning about the volume of a cone.
Informal Argument for the Volume of a Cone
The volume \( V \) of a cylinder is given by the formula: \[ V_{cylinder} = \pi r^2 h \] where \( r \) is the radius and \( h \) is the height. A cone with the same base radius and height, when put into context, can be thought of as tapering off to a point rather than extending completely to the top.
To understand how a cone relates to a cylinder, we can visually think about dividing the cylinder into smaller pieces. If we fill a cone with a similar base and height as a cylinder, the cone takes up less space. By geometric reasoning and calculus, it can be shown that a cone's volume is precisely one-third of that of a cylinder with the same base and height.
Thus, to find the volume of a cone that has the same height and base radius as a cylinder, we say:
- Answer: a. You need to multiply the volume of the cylinder by \( \frac{1}{3} \).
Volume of Eduardo's Cone
Eduardo’s cone has a diameter of 30 cm, so the radius \( r \) is: \[ r = \frac{diameter}{2} = \frac{30}{2} = 15 , \text{cm} \] The height \( h \) is twice the diameter, thus: \[ h = 2 \times 30 = 60 , \text{cm} \]
The volume \( V \) of a cone is given by: \[ V = \frac{1}{3} \pi r^2 h \] Substituting in the values: \[ V = \frac{1}{3} \pi (15)^2 (60) = \frac{1}{3} \pi (225)(60) = \frac{1}{3} \pi (13500) = 4500\pi , \text{cm}^3 \]
- Answer: b. \( 4,500\pi , \text{cm}^3 \)
Height of the New Ice Cream Cone
The volume of the original cone with radius 3 inches and height 6 inches is: \[ V = \frac{1}{3} \pi (3^2)(6) = \frac{1}{3} \pi (9)(6) = \frac{1}{3} \pi (54) = 18\pi , \text{in}^3 \]
For the wider cone with a radius of 4 inches, we set the height \( h' \) in the volume formula: \[ V = \frac{1}{3} \pi (4^2)(h') \] Setting the volumes equal: \[ 18\pi = \frac{1}{3} \pi (16)(h') \] Cancel \( \pi \) and multiply by 3: \[ 54 = 16h' \] \[ h' = \frac{54}{16} = 3.375 \]
Rounding to the nearest inch gives:
- Answer: a. 3 inches
Volume of a Cone with Given Dimensions
Given a cone with a height of 9 inches and a diameter of 4 inches, the radius \( r \) is: \[ r = \frac{diameter}{2} = \frac{4}{2} = 2 , \text{in} \]
Now, substitute into the volume formula: \[ V = \frac{1}{3} \pi (2^2)(9) = \frac{1}{3} \pi (4)(9) = \frac{1}{3} \pi (36) = 12\pi , \text{in}^3 \]
- Answer: a. \( 12\pi , \text{in}^3 \)
Volume of Tarik’s Cone
To determine the volume accurately, we would need the measurements of Tarik's cone. Assuming each answer represents a possible solution to a volume calculation, without additional dimensions, here are the options provided and a general understanding of how we calculate:
If we had the base radius \( r \) and height \( h \) or necessary information, we could use: \[ V = \frac{1}{3} \pi r^2 h \]
Without specific measurements, it's impossible to verify Tarik's volume. If dimensions were provided, the calculation would determine which option fits.
If we were to assume a value, nothing can be calculated here rightly without dimensions. Hence, the accurate response cannot be given.
If you can provide those details, the solution will be resolvable with the volume formula again.
However, as per the options list, the closest intuitive guess without clear information could be evaluated:
- [Options for Volume of the Cone]: Without dimensions, solving accurately cannot be done. Please provide dimensions for a precise answer.
So feel free to correct or supply additional details, and I can help further!