The volume of the rectangular prism is 32 cubic feet more than the volume of the right triangular prism. Find the volume of each

Answer:

Step-by-step explanation:

Denote the rectangular prism's volume as VR and the volume of the right triangular prism as VT. Given that the rectangular prism's volume exceeds that of the triangular prism by 32 cubic feet, we establish that VR = 32 + VT.

Calculating VR: VR = length * width * height = 6*x*3.

This simplifies to VR = 18x ft³.

For the triangular prism, VT can be calculated as Length * width * Height / 2, yielding VT = (7 * x * 4) / 2.

Thus, VT = 28x / 2, which simplifies to VT = 14x ft³.

Equating VR to 32 + VT allows the substitution: 18x = 32 + (14x).

Now, simplify the equation:

Combine like terms: 18x - 14x = 32.

This results in 4x = 32.

Dividing both sides by 4:

x = 8.

The volume for the rectangular prism comes to 18x, which equals 18*8.

The volume of the rectangular prism calculates to 144 ft³.

The volume for the right triangular prism calculates to 14x, which equals 14*8.

The volume of the right triangular prism calculates to 112 ft³.

Answer:

The rectangular prism has a volume of 144 ft³ and the triangular prism has a volume of 112 ft³.

Step-by-step explanation:

Let the volume of the rectangular prism be known as VR and the volume of the right triangular prism as VT. It is established that VR = 32 + VT, indicating that the rectangular prism holds 32 cubic feet more than the triangular prism.

Calculating for the rectangular prism: VR = length * width * height yields VR = 6*x*3.

This results in VR equaling 18x ft³.

Next, for the triangular prism, VT = Length * width * Height / 2 results in VT = (7 * x * 4) / 2.

This becomes VT = 28x / 2 or 14x ft³.

Using the relationship VR = 32 + VT gives us the equation: 18x = 32 + (14x).

Combining similar terms:

This gives 18x - 14x = 32.

The simplified result is 4x = 32.

Dividing both sides by 4 leads us to:

x = 8.

Therefore, the rectangular prism's volume is 18x, which calculates to 18*8.

The finalized volume of the rectangular prism is 144 ft³.

The triangular prism's volume is computed as 14x, where x = 8, resulting in 14*8.

Thus, the triangular prism's volume is 112 ft³.