The shape, a cube and a square based pyramid. Both shapes have the same base and the same height. Which expression represents the total volume of this shape? justify your answer.

Note:that the pyramid is on top of the cube joined together.

volume of cube = base x height

volume of pyramid = (1/3)base x height

total volume = base x height + (1/3)base x height
= (4/3)base x height

thanks.

To find the total volume of the shape, we need to calculate the volumes of both the cube and the square-based pyramid.

Let's start by determining the volume of the cube. The volume of a cube is given by the formula V_cube = side^3, where "side" represents the length of one side of the cube.

Since the base of the square-based pyramid is the same as the base of the cube, its side length will also be the same. So, we can represent the volume of the cube as V_cube = side^3.

Moving on to the square-based pyramid, the volume formula is V_pyramid = (base area * height) / 3. Since the base of the pyramid is a square, the base area is calculated by multiplying the length of one side by itself, which can be written as base area = side^2.

As mentioned earlier, the height of the pyramid is the same as the side length of the cube, so we can represent it as height = side.

Substituting these values into the volume formula for the pyramid, we get V_pyramid = (side^2 * side) / 3.

To find the total volume of the combined shape (cube + pyramid), we need to add the volume of the cube and the volume of the pyramid: Total volume = V_cube + V_pyramid.

Substituting the expressions we derived earlier, we have:
Total volume = side^3 + (side^2 * side) / 3.

So, the expression that represents the total volume of this shape is side^3 + (side^2 * side) / 3.

This expression can be justified by considering the individual volume formulas for both the cube and pyramid, and adding them together to find the total volume of the combined shape.