A rectangular prism has a base of 42 square inches. If the volume of the prism is 210 cubic inches, which could be dimensions of this prism
I dont think this was helpful
Volume = area of base x height
210 = 42 h
h = 5
Of course there is an infinite number of possible dimensions. If we are restricted to whole numbers, possible cases could be
6 x 7 x 5 , or
2 x 21 x 5, or
3 x 14 x 5, or
1 x 42 x 5 or
1 x 1 x 210 , or
7 x 3 x 10 , or .....
To find the dimensions of the prism, we need to consider that the volume of a rectangular prism is given by multiplying its length, width, and height.
Let's assume the dimensions of the prism to be length (L), width (W), and height (H).
Given that the base of the prism has an area of 42 square inches, we can write:
L * W = 42 (Equation 1)
Also, the volume of the prism is given as 210 cubic inches:
L * W * H = 210 (Equation 2)
From Equation 1, we can see that the length (L) and width (W) must be factors of 42. Let's find all the possible pairs of values for (L, W) that satisfy this condition:
Pair 1: (L = 1, W = 42)
Pair 2: (L = 2, W = 21)
Pair 3: (L = 3, W = 14)
Pair 4: (L = 6, W = 7)
Pair 5: (L = 7, W = 6)
Pair 6: (L = 14, W = 3)
Pair 7: (L = 21, W = 2)
Pair 8: (L = 42, W = 1)
Now, let's substitute these pairs into Equation 2 to find the corresponding values of H:
For Pair 1: L * W * H = 1 * 42 * H = 42H
For Pair 2: L * W * H = 2 * 21 * H = 42H
For Pair 3: L * W * H = 3 * 14 * H = 42H
For Pair 4: L * W * H = 6 * 7 * H = 42H
For Pair 5: L * W * H = 7 * 6 * H = 42H
For Pair 6: L * W * H = 14 * 3 * H = 42H
For Pair 7: L * W * H = 21 * 2 * H = 42H
For Pair 8: L * W * H = 42 * 1 * H = 42H
From the above calculations, we can see that for any value of L and W that satisfy Equation 1, the height H must be such that L * W * H = 42H = 210, which simplifies to H = 5.
Therefore, the possible dimensions of the rectangular prism could be:
1) Length = 1 inch, Width = 42 inches, Height = 5 inches
2) Length = 2 inches, Width = 21 inches, Height = 5 inches
3) Length = 3 inches, Width = 14 inches, Height = 5 inches
4) Length = 6 inches, Width = 7 inches, Height = 5 inches
5) Length = 7 inches, Width = 6 inches, Height = 5 inches
6) Length = 14 inches, Width = 3 inches, Height = 5 inches
7) Length = 21 inches, Width = 2 inches, Height = 5 inches
8) Length = 42 inches, Width = 1 inch, Height = 5 inches
To find the dimensions of the rectangular prism, we need to consider the formula for the volume of a rectangular prism: Volume = length × width × height.
In this case, we are given the volume (210 cubic inches) and the base area (42 square inches). The base area is equal to the length multiplied by the width.
Let's assume the length of the rectangular prism is L, the width is W, and the height is H.
We have two equations:
1) L × W = 42 (Equation 1)
2) L × W × H = 210 (Equation 2)
We can solve Equation 1 for L:
L = 42 / W
Substituting L in Equation 2, we get:
(42 / W) × W × H = 210
42H = 210
H = 210 / 42
H = 5
Now, we can substitute H = 5 back into Equation 2:
L × W × 5 = 210
L × W = 210 / 5
L × W = 42
Comparing this equation with Equation 1, we observe that they are the same. Therefore, it is not possible to determine the exact values of length and width independently. We have an infinite number of possible solutions.
For example, if we assume the length is 6 and the width is 7, the equations are satisfied:
L = 6
W = 7
H = 5
L × W = 6 × 7 = 42
42 × 5 = 210
Thus, one possible set of dimensions for the prism is 6 inches × 7 inches × 5 inches. However, there could be other valid combinations of length and width that satisfy the given conditions.