Answer:
1.38 m³
Step-by-step explanation:
The volume of the treasure chest can be calculated by summing the volume of a rectangular prism and the volume of a half-cylinder.
The formula for the volume of a rectangular prism is:
[tex]\boxed{\begin{array}{l}\underline{\textsf{Volume of a rectangular prism}}\\\\V=l \times w \times h\\\\\textsf{where:}\\\phantom{ww}\bullet\;\textsf{$V$ is the volume.}\\\phantom{ww}\bullet\;\textsf{$l$ is the length of the base.}\\\phantom{ww}\bullet\;\textsf{$w$ is the width of the base.}\\\phantom{ww}\bullet\;\textsf{$h$ is the height.}\end{array}}[/tex]
In this case:
- l = 0.8 m
- w = 0.7 m
- h = 1.7 m
Substitute the values into the formula and solve for V:
[tex]V_{\text{rectangular prism}}=0.8 \times 0.7 \times 1.7\\\\V_{\text{rectangular prism}}=0.952\; \rm m^3[/tex]
The formula for the volume of a half-cylinder is:
[tex]\boxed{\begin{array}{l}\underline{\textsf{Volume of a half-cylinder}}\\\\V=\dfrac{\pi r^2 h}{2}\\\\\textsf{where:}\\\phantom{ww}\bullet\;\textsf{$V$ is the volume.}\\\phantom{ww}\bullet\;\textsf{$r$ is the radius of the circular base.}\\\phantom{ww}\bullet\;\textsf{$h$ is the height.}\end{array}}[/tex]
In this case, the diameter of the circular base is 0.8 meters. Since the radius of a circle is half its diameter, r = 0.4 m. The height of the cylinder is h = 1.7 m. Therefore:
[tex]V_{\text{half-cylinder}}=\dfrac{\pi \times 0.4^2 \times 1.7}{2}\\\\\\V_{\text{half-cylinder}}=\dfrac{\pi \times 0.16 \times 1.7}{2}\\\\\\V_{\text{half-cylinder}}=\dfrac{0.272\pi}{2}\\\\\\V_{\text{half-cylinder}}=0.136\pi\; \rm m^3[/tex]
To find the total volume, add together the two volumes:
[tex]\text{Total volume}=V_{\text{rectangular prism}}+V_{\text{half-cylinder}}\\\\\text{Total volume}=0.952+0.136\pi\\\\\text{Total volume}=0.952+0.42725660...\\\\\text{Total volume}=1.37925660...\\\\\text{Total volume}=1.38\; \rm m^3\;(2\;d.p.)[/tex]
Therefore, the total volume of the treasure chest rounded to two decimal places is:
[tex]\LARGE\boxed{\boxed{1.38\; \rm m^3}}[/tex]
