I need to find the dimensions of this rectangular prism. It has a volume of 108 cubic inches. The length and width of the base are identical. The height is half of the length. Can you help explain how to do this given this information?
V = L * H * W
L = W = 2H
V = 2H * H * 2H
Insert values and solve for H, then L and W.
Sure! To find the dimensions of the rectangular prism, we need to use the given information about the volume and relationship between the length, width, and height.
Let's start by identifying what we know:
1. The volume of the rectangular prism is 108 cubic inches.
2. The length and width of the base are identical.
3. The height is half of the length.
Now, let's set up the equations based on the information given:
1. The volume of a rectangular prism is calculated by multiplying the length, width, and height. So we can write the equation as: length * width * height = 108.
2. From the second piece of information, we know that the length and width are identical. Let's call this value x. So, the equation becomes: x * x * height = 108.
3. According to the third piece of information, the height is half of the length. So, the equation becomes: x * x * (0.5x) = 108.
Now, let's solve the equation to find the value of x:
1. Simplify the equation: x^2 * (0.5x) = 108.
2. Multiply: 0.5x^3 = 108.
3. Divide both sides by 0.5: x^3 = 216.
4. Take the cube root of both sides: x = ∛216.
5. Simplify: x = 6.
Now that we have the value of x, we can find the dimensions of the rectangular prism:
- Length = x = 6.
- Width = x = 6.
- Height = 0.5x = 0.5 * 6 = 3.
Therefore, the dimensions of the rectangular prism are: 6 inches in length, 6 inches in width, and 3 inches in height.
Yes, I can help you with that. To find the dimensions of the rectangular prism, let's use the given information step by step.
Step 1: Determine the relationship between the length, width, and height:
You mentioned that the length and width of the base are identical. Let's call this common value "x". The height is described as half of the length, so the height will be "x/2".
Step 2: Write the volume formula for a rectangular prism:
The volume of a rectangular prism is calculated by multiplying the length, width, and height. So, the volume formula is: volume = length × width × height.
Step 3: Substitute the given values into the volume formula:
We are given that the volume is 108 cubic inches. So, we can write the equation as:
108 = x × x × (x/2).
Step 4: Solve the equation for x:
To simplify the equation, we can multiply the x's together:
108 = x² × (x/2).
We can simplify further by canceling out the common factor of 2:
108 = (x² × x) / 2.
Multiplying both sides by 2:
216 = x³.
To find the value of x, we take the cube root of both sides:
∛216 = ∛x³.
6 = x.
Step 5: Determine the dimensions of the rectangular prism:
We found that x = 6, which represents the length and width of the base. Since the height is half the length, the height is x/2, which is 6/2 = 3.
Therefore, the dimensions of the rectangular prism are:
Length = Width = 6 inches,
Height = 3 inches.
So, the rectangular prism has dimensions of 6 inches in length and width, and 3 inches in height.