Fruit cocktail comes in large and small cylindrical cans. The larger can has a radius and height that are both twice as long as that of the radius and height of the smaller can. If the volume of the smaller can is $28.64 in ^3$, what is the volume of the larger can?
A. 57.28 in $^3$
B. $114.56 in ^3$
C. $229.12 in ^3$
D. $171.84 in ^3$
Answer (2)
Define the radius and height of the smaller can as r s and h s , respectively, and express its volume as V s = π r s 2 h s = 28.64 in 3 .
Express the radius and height of the larger can as r l = 2 r s and h l = 2 h s , respectively.
Determine the volume of the larger can as V l = π r l 2 h l = 8 π r s 2 h s = 8 V s .
Calculate the volume of the larger can: V l = 8 × 28.64 = 229.12 in 3 .
Explanation
Problem Analysis Let's analyze the problem. We have two cylindrical cans, a smaller one and a larger one. The dimensions (radius and height) of the larger can are twice the dimensions of the smaller can. We know the volume of the smaller can and need to find the volume of the larger can.
Volume of Smaller Can Let r s and h s be the radius and height of the smaller can, respectively. Then, the volume of the smaller can, V s , is given by the formula: V s = π r s 2 h s We are given that V s = 28.64 in 3 .
Volume of Larger Can The radius and height of the larger can are twice those of the smaller can. So, the radius of the larger can is r l = 2 r s and the height of the larger can is h l = 2 h s . The volume of the larger can, V l , is given by: V l = π r l 2 h l = π ( 2 r s ) 2 ( 2 h s ) V l = π ( 4 r s 2 ) ( 2 h s ) = 8 π r s 2 h s = 8 V s
Calculate Volume of Larger Can Since V s = 28.64 in 3 , we can substitute this value into the equation for V l :
V l = 8 V s = 8 ( 28.64 ) = 229.12 in 3
Final Answer Therefore, the volume of the larger can is 229.12 in 3 .
Examples
Cylindrical cans are commonly used for packaging various food items. Knowing how the volume changes with the dimensions is useful in designing packaging. For example, if you want to double the amount of product in a can while keeping the same proportions, you can calculate the new dimensions needed. This is also applicable in other scenarios like designing storage tanks or containers where volume calculations are important.
The volume of the larger can is calculated to be 229.12 in³ by using the relationship between the dimensions of the smaller and larger cans. The larger can's dimensions are twice those of the smaller can, leading to a volume that is eight times greater than the smaller can's volume. Thus, the correct answer is option C.
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