To obtain the volume of the solid figure, the following steps are necessary:
Step 1: Since the figure is made up of a square base pyramid and a rectangular solid, we recall the formulas for the volumes of a square base pyramid and that of a rectangular solid, as below:
[tex]V_{square\text{ pyramid}}=\frac{1}{3}\times base\text{ area}\times perpendicular\text{ height}[/tex]
And:
[tex]V_{rec\tan gular\text{ solid}}=length\times width\times height[/tex]
Step 2: Apply the dimensions given in the question to obtain the volumes, as follows:
Consider the rectangular solid sketched out below:
Thus, for the rectangular solid, we have that:
Length = 9in
width = 9in
height = 16in
Therefore, the volume of the rectngular solid is:
[tex]\begin{gathered} V_{rec\tan gular\text{ solid}}=length\times width\times height \\ V_{rec\tan gular\text{ solid}}=9in\times9in\times16in \\ V_{rec\tan gular\text{ solid}}=1296in^3 \end{gathered}[/tex]
Now, consider the square base pyramid sketched out below:
Thus, for the square base pyramid, we have that:
[tex]\text{base area = 9in}\times9in=81in^2[/tex]
Also:
[tex]\text{perpendicular height = (26in-16in)= 10in}[/tex]
Therefore, the volume of the square base pyramid is:
[tex]\begin{gathered} V_{square\text{ pyramid}}=\frac{1}{3}\times base\text{ area}\times perpendicular\text{ height} \\ V_{square\text{ pyramid}}=\frac{1}{3}\times81in^2\times10in \\ V_{square\text{ pyramid}}=\frac{810}{3}in^3 \\ V_{square\text{ pyramid}}=270in^3 \end{gathered}[/tex]
Step 3: Now, we find the total volume of the solid figure as follows:
[tex]V_{solid\text{ figure}}=V_{rec\tan gular\text{ solid}}+V_{square\text{ pyramid}}[/tex]
Since:
[tex]V_{rec\tan gular\text{ solid}}=1296in^3[/tex]
And:
[tex]V_{square\text{ pyramid}}=270in^3[/tex]
Therefore:
[tex]\begin{gathered} V_{solid\text{ figure}}=V_{rec\tan gular\text{ solid}}+V_{square\text{ pyramid}} \\ V_{solid\text{ figure}}=1296_{}+270 \\ V_{solid\text{ figure}}=1566in^3 \end{gathered}[/tex]
Thus, the volume of the solid figure is 1566 cubic inches
