If each edge of a cube is increased by 2 inches, the
A. volume is increased by 8 cubic inches
B. area of each face is increased by 4 square inches
C. diagonal of each face is increased by 2 inches
D. sum of the edges is increased by 24 inches
is the answer C ?
how many edges does a cube have?
a cube has 12 edges, so would the answer be D?
yes
To find the correct answer, let's go through the options one by one and see how increasing each edge of a cube by 2 inches affects different measurements:
A. Volume is increased by 8 cubic inches: The volume of a cube is calculated by the formula V = s^3, where s is the length of one edge. By increasing each edge by 2 inches, the new edge length would be (s + 2). So, the new volume would be (s + 2)^3. Let's check if the increase in volume is 8 cubic inches:
(s + 2)^3 - s^3 = 8
Expanding the equation:
(s^3 + 6s^2 + 12s + 8) - s^3 = 8
The s^3 terms cancel out:
6s^2 + 12s + 8 = 8
Simplifying:
6s^2 + 12s = 0
This equation does not result in a value of s that would satisfy the given condition. So, we can eliminate option A.
B. Area of each face is increased by 4 square inches: The area of one face of the cube is calculated by the formula A = s^2, where s is the length of one edge. By increasing each edge by 2 inches, the new edge length would be (s + 2). So, the new area would be (s + 2)^2. Let's check if the increase in area is 4 square inches:
(s + 2)^2 - s^2 = 4
Expanding the equation:
(s^2 + 4s + 4) - s^2 = 4
The s^2 terms cancel out:
4s + 4 = 4
Simplifying:
4s = 0
This equation also does not result in a value of s that would satisfy the given condition. So, we can eliminate option B.
C. Diagonal of each face is increased by 2 inches: The diagonal of a face of the cube is equal to the length of the edge times the square root of 2. If we increase each edge by 2 inches, the new edge length would be (s + 2). So, the new diagonal would be (s + 2) * √2. Let's check if the increase in diagonal is 2 inches:
(s + 2) * √2 - s * √2 = 2
√2s + 2√2 - √2s = 2
2√2 = 2
This equation satisfies the condition, indicating that the length of the diagonal of each face is indeed increased by 2 inches. Therefore, option C is correct.
D. Sum of the edges is increased by 24 inches: The sum of the edges of a cube is calculated by the formula S = 12s, where s is the length of one edge. By increasing each edge by 2 inches, the new edge length would be (s + 2). So, the new sum of edges would be 12 * (s + 2). Let's check if the increase in sum of edges is 24 inches:
12 * (s + 2) - 12s = 24
12s + 24 - 12s = 24
24 = 24
This equation satisfies the condition, indicating that the sum of the edges is indeed increased by 24 inches. Therefore, option D is correct as well.
In conclusion, both options C and D are correct answers to the question.