A rectangular prism has a high of 3 inches and a volume of 27 in³. the height of the prism is changed to 6 inches and the other dimension stay the same. what is the volume of the prism with this change?
since the change is in only one of the dimensions, the change of the volume must be linear.
height was tripled, so the volume is doubled or
new volume is 54 in^3
or
old volume = 3(l)(w) = 3lw = 27
lw = 9
new volume = 6(l)(w) = 6lw = 6(9) = 54
To find the new volume of the rectangular prism, we know that the height is changed from 3 inches to 6 inches, while the other dimensions stay the same.
Let's denote the original dimensions as length (L), width (W), and height (H), and the new dimensions as length (L), width (W), and new height (H').
Given that the original volume (V) is 27 in³, and the original height (H) is 3 inches, we can use the formula for the volume of a rectangular prism:
V = L * W * H
Substituting the given values, we have:
27 = L * W * 3
We want to find the new volume (V'), using the new height (H'), which is 6 inches. The formula for volume remains the same, so we have:
V' = L * W * H'
Since the length (L) and width (W) remain unchanged, we can write:
V' = V * (H'/H)
Substituting the given values, we get:
V' = 27 * (6/3)
Simplifying this equation, we find:
V' = 27 * 2
V' = 54
Therefore, the new volume of the rectangular prism with the changed height is 54 in³.
To find the volume of the rectangular prism after the change, we need to determine the new dimensions. Since we know the original volume and the new height, we can use the formula for volume to solve for the missing dimension.
The formula for the volume of a rectangular prism is:
Volume = Length x Width x Height
Given that the original height is 3 inches and the original volume is 27 in³, we can substitute these values into the formula:
27 = Length x Width x 3
Now we need to solve for the missing dimensions, Length and Width. Since the problem states that the other dimension remains the same, we know that only the height changes from 3 inches to 6 inches.
Let's solve for Length and Width using the formula:
27 = Length x Width x 3
Dividing both sides of the equation by 3, we get:
9 = Length x Width
Now, let's consider the new height of 6 inches. The new volume can be found by substituting the new height into the formula:
New Volume = Length x Width x 6
Substituting the value of Length x Width (which we found to be 9) into the formula, we have:
New Volume = 9 x 6 = 54 in³
Therefore, the volume of the prism after the change in height is 54 cubic inches.