The volume of a rectangular box is $2 x(2 x+4)(2 x-2)$. (Drawing is not to scale.)
Which statement about the volume of the box is true?
A. The volume is the product of the length, $2 x+4$, and the width, $2 x$
B. The volume does not depend on the height, $2 x-2$.
C. The volume is the sum of the length, $2 x+4$, the width, $2 x$, and the height, $2 x-2$
D. The volume is the product of the area of the base, $2 x(2 x+4)$, and the height, $2 x-2
Answer (1)
The volume of the box is given as 2 x ( 2 x + 4 ) ( 2 x − 2 ) .
Statement A is incorrect because it only considers the product of two dimensions.
Statement B is incorrect because the volume clearly depends on the height ( 2 x − 2 ) .
Statement C is incorrect because the volume is a product, not a sum, of the dimensions.
Statement D correctly states that the volume is the product of the base area 2 x ( 2 x + 4 ) and the height ( 2 x − 2 ) , so the answer is D .
Explanation
Analyze each statement The volume of a rectangular box is given as 2 x ( 2 x + 4 ) ( 2 x − 2 ) . We need to determine which statement about the volume is true. Let's analyze each option:
A. The volume is the product of the length, 2 x + 4 , and the width, 2 x This would mean the volume is ( 2 x ) ( 2 x + 4 ) = 4 x 2 + 8 x . This is not equal to the given volume 2 x ( 2 x + 4 ) ( 2 x − 2 ) .
B. The volume does not depend on the height, 2 x − 2 This is false because the volume expression 2 x ( 2 x + 4 ) ( 2 x − 2 ) clearly includes the term ( 2 x − 2 ) .
C. The volume is the sum of the length, 2 x + 4 , the width, 2 x , and the height, 2 x − 2 This would mean the volume is ( 2 x + 4 ) + ( 2 x ) + ( 2 x − 2 ) = 6 x + 2 . This is not equal to the given volume 2 x ( 2 x + 4 ) ( 2 x − 2 ) .
D. The volume is the product of the area of the base, 2 x ( 2 x + 4 ) , and the height, 2 x − 2 This would mean the volume is [ 2 x ( 2 x + 4 )] ( 2 x − 2 ) = 2 x ( 2 x + 4 ) ( 2 x − 2 ) , which matches the given volume.
Therefore, the correct statement is D.
Verify each statement The volume of the rectangular box is given by the expression 2 x ( 2 x + 4 ) ( 2 x − 2 ) . We need to verify which of the given statements is true.
Statement A suggests the volume is simply the product of 2 x and 2 x + 4 . However, the actual volume also includes the factor ( 2 x − 2 ) , so statement A is incorrect.
Statement B claims the volume doesn't depend on ( 2 x − 2 ) . This is false because ( 2 x − 2 ) is a factor in the volume expression.
Statement C suggests the volume is the sum of 2 x , 2 x + 4 , and 2 x − 2 . This is also incorrect because the volume is a product, not a sum, of these terms.
Statement D states that the volume is the product of the base area 2 x ( 2 x + 4 ) and the height ( 2 x − 2 ) . This matches the given volume expression 2 x ( 2 x + 4 ) ( 2 x − 2 ) .
Thus, statement D is the correct one.
Determine the correct statement The volume of a rectangular box is given by 2 x ( 2 x + 4 ) ( 2 x − 2 ) . We need to determine which of the given statements is true.
Statement A: The volume is the product of the length, 2 x + 4 , and the width, 2 x . This is incorrect because it doesn't include the height, 2 x − 2 .
Statement B: The volume does not depend on the height, 2 x − 2 . This is false because the volume expression includes the term ( 2 x − 2 ) .
Statement C: The volume is the sum of the length, 2 x + 4 , the width, 2 x , and the height, 2 x − 2 . This is incorrect because the volume is a product, not a sum.
Statement D: The volume is the product of the area of the base, 2 x ( 2 x + 4 ) , and the height, 2 x − 2 . This is correct because 2 x ( 2 x + 4 ) represents the area of the base and ( 2 x − 2 ) represents the height, and their product gives the volume.
Therefore, the correct statement is D.
Final Answer The correct statement is D.
Examples
Understanding the volume of a rectangular box is crucial in various real-life scenarios. For instance, when packing items into a shipping container, knowing the volume helps determine how many boxes can fit inside. Similarly, in construction, calculating the volume of concrete needed for a foundation ensures accurate material ordering and cost estimation. This concept also applies to everyday tasks like organizing storage spaces or estimating the amount of water needed to fill an aquarium.
