Question

a cube of ( 5\times5\times5 ) is shown in the figure below. the rows and columns of smaller cubes which are indicated with shading have been removed. the volume of this solid figure is ____.

Explanation:

Step1: Calculate the volume of the original cube

The volume formula of a cube is \(V = a^3\) (where \(a\) is the side - length). For a cube with \(a = 5\), the volume of the original cube \(V_{original}=5\times5\times5=125\).

Step2: Count the number of removed small cubes

• First layer (top - view):
• The number of removed small cubes in the top - view (shaded row and column) is \(5 + 5-1=9\) (we subtract 1 to avoid double - counting the intersection cube).
• Second layer (front - view, non - overlapping with the first - layer removal in the vertical direction):
• There are 3 removed small cubes.
• Third layer (back - view, non - overlapping with the first - layer removal in the vertical direction):
• There are 3 removed small cubes.
• The total number of removed small cubes \(n=9 + 3+3=15\).
• Since the volume of each small cube is \(1\times1\times1 = 1\).

Step3: Calculate the volume of the remaining solid

The volume of the remaining solid \(V=V_{original}-n\). Substitute \(V_{original}=125\) and \(n = 15\) into the formula, we get \(V=125-15=110\).

Answer:

110