Final answer:
The volume of shape Q is determined using the ratio of surface areas and the principle that the ratio of volumes of similar shapes is the cube of their linear dimensions ratio. This yields a volume of 21600 cm³ for shape Q.
Step-by-step explanation:
The student's question revolves around finding the volume of a geometrically similar shape Q given the surface areas and volume of shape P. To find the volume of shape Q, we utilize the principle that the ratio of the volumes of two similar shapes is the cube of the ratio of their corresponding lengths. This ratio can be inferred from the ratio of their surface areas since the surface area scales with the square of the ratio of corresponding lengths.
The given total surface area of shape P is 540 cm² and that of shape Q is 2160 cm². We can determine this ratio as follows:
Surface Area Ratio (Q to P) = Surface Area of Q / Surface Area of P = 2160 cm² / 540 cm² = 4
Since surface areas are proportional to the square of their corresponding lengths, the ratio of the volumes will be the cube of the square root of the surface area ratio:
Volume Ratio (Q to P) = (Surface Area Ratio)^(3/2) = 4^(3/2) = 8
Now, knowing the volume of shape P, we can find the volume of shape Q:
Volume of Q = Volume of P × Volume Ratio = 2700 cm³ × 8 = 21600 cm³
Hence, the volume of shape Q is 21600 cm³.