A cone has a radius of 40 cm and a volume of 1,875 cm³. What is the volume of a similar cone with a radius of 16 cm?
A.) 120 cm³
B.) 300 cm³
C.) 75 cm³
D.) 750 cm³
C.) 75 cm³
1875 * (16^3 / 40^3) = 120 .... A.)
To find the volume of a similar cone, we can use the concept of ratios.
The ratio of the radii of the two cones is 16 cm / 40 cm, which simplifies to 2/5.
Since the volume of a cone is directly proportional to the cube of its radius, we can say that the ratio of the volumes of the two cones is equal to the cube of the ratio of their radii.
So, (volume of smaller cone) / (volume of larger cone) = (2/5)³
Now, we can solve for the volume of the smaller cone:
(volume of smaller cone) = (volume of larger cone) * ((2/5)³)
Given that the volume of the larger cone is 1,875 cm³, we can substitute this value into the equation:
(volume of smaller cone) = 1,875 cm³ * ((2/5)³)
(volume of smaller cone) = 1,875 cm³ * (8/125)
(volume of smaller cone) = 150,000 / 125
(volume of smaller cone) = 1,200 cm³
Therefore, the volume of the similar cone with a radius of 16 cm is 1,200 cm³.
So, the correct answer is A.) 120 cm³.
To find the volume of a similar cone, we need to understand the relationship between similar shapes. When two shapes are similar, their corresponding sides are proportional to each other.
In this case, the ratio of the radii of the two cones is 16 cm / 40 cm = 0.4. Since volume is a three-dimensional measure, it is proportional to the cube of the lengths.
Therefore, the ratio of the volumes of the two cones is (0.4)³ = 0.064.
To find the volume of the similar cone with a radius of 16 cm, we can multiply the volume of the original cone by the ratio we found.
1,875 cm³ * 0.064 = 120 cm³.
Therefore, the volume of the similar cone with a radius of 16 cm is 120 cm³.
Hence, the correct option is A.) 120 cm³.