Final answer:
To find the volume of the treasure chest, calculate the volume of the rectangular part and the semicircular part separately and then add them together. The total volume is the sum of the volume of the rectangle and the volume of the semicircle, multiplied by the length of the chest.
Step-by-step explanation:
To find the volume of a treasure chest that is shaped like a prism with a semicircular cross-section, we must calculate the volume of the rectangular part and the semicircular part separately and then add them together. The base of the rectangular part of the prism is 0.8 units, the height is 0.9 units, and the length is 1.9 units. The formula for the volume of a rectangle is V = lwh where l is length, w is width, and h is height. The volume of the rectangular part is thus Vrect = 1.9 × 0.8 × 0.9.
For the semicircular part, we first calculate the area of a full circle and then halve it. The formula for the area of a circle is A = πr². In this case, the diameter of the semicircle equals the width (base) of the rectangle, which is 0.8 units, so the radius r is 0.4 units. Hence, the area of the semicircle is Asemi = ½ π (0.4)² and therefore the volume of the semicircular part is Vsemi = ½ π (0.4)² × 1.9. Finally, we add both volumes:
Total Volume = Vrect + Vsemi = (1.9 × 0.8 × 0.9) + (½ π (0.4)² × 1.9).