Let's go through each question one by one:
Question 1
A) For the volume of a cone, an informal argument is that a cone can be thought of as a pyramid with a circular base. The formula for the volume of a cylinder is V = πr²h, where r is the radius and h is the height. Since a cone takes up one third of the space in the cylinder with the same base and height, the volume of the cone is given by V = (1/3)πr²h.
The correct response is: You need to multiply the volume of the cylinder by \( \frac{1}{3} \).
Question 2
A) The diameter of the cone is 30 cm, so the radius \( r \) is half of that: \( r = 15 \) cm. The height \( h \) is twice the diameter, which gives us a height of \( h = 30 \) cm.
Using the volume formula for a cone \( V = \frac{1}{3}πr^2h \):
- \( V = \frac{1}{3}π(15)^2(30) = \frac{1}{3}π(225)(30) = \frac{1}{3}π(6750) = 2250π , \text{cm}^3 \).
The correct response is: 2,250π cm³.
Question 3
A) The volume of the original cone is calculated with \( r = 3 \) inches and \( h = 6 \) inches:
- \( V = \frac{1}{3}π(3)^2(6) = \frac{1}{3}π(9)(6) = \frac{1}{3}π(54) = 18π \) cubic inches.
In the new cone, the radius \( r = 4 \) inches and we need to find the new height \( h' \) such that the volume remains the same:
- \( V = \frac{1}{3}π(4)^2h' \Rightarrow 18π = \frac{1}{3}π(16)h' \).
Cancelling \( π \) and multiplying both sides by 3:
- \( 54 = 16h' \Rightarrow h' = \frac{54}{16} = 3.375 \) inches.
Rounding to the nearest inch:
- This is approximately \( 3 \) inches.
The correct response is: 3 inches.
Question 4
A) The cone has a height of 9 inches and a diameter of 4 inches, so the radius \( r = 2 \) inches. Using the volume formula:
- \( V = \frac{1}{3}π(2)^2(9) = \frac{1}{3}π(4)(9) = \frac{1}{3}π(36) = 12π \) cubic inches.
The correct response is: 12π in³.
Question 5
A) The cone has a height of 11 and a radius of 6: Using the volume formula:
- \( V = \frac{1}{3}π(6)^2(11) = \frac{1}{3}π(36)(11) = \frac{1}{3}π(396) = 132π \) cubic units.
The correct response is: 132π cubic units.