Answer: a) To find the scale factor, we need to find the ratio of the corresponding lengths of the two prisms, since the volumes of two similar prisms are proportional to the cube of the scale factor. Let x be the scale factor.
(volume of larger prism) / (volume of smaller prism) = (x³) = 27/1
x³ = 27
x = 3
Therefore, the scale factor is 3.
b) The ratio of the areas is equal to the square of the scale factor. Let A1 and A2 be the areas of the larger and smaller prisms, respectively.
(A1) / (A2) = (scale factor)²
A2 = A1 / (scale factor)²
A2 = 153 / 9
A2 = 17 cm²
Therefore, the ratio of their areas is 9:1.
c) The ratio of the volumes is equal to the cube of the scale factor.
(volume of larger prism) / (volume of smaller prism) = (scale factor)³
(volume of smaller prism) = (1 cm³) * (scale factor)³
(volume of smaller prism) = 27 cm³
Therefore, the ratio of their volumes is 27:1.
d) To find the surface area of the smaller prism, we can use the same ratio as in part (b) to find that the surface area of the smaller prism is equal to the surface area of the larger prism divided by the square of the scale factor.
(surface area of smaller prism) = (surface area of larger prism) / (scale factor)²
(surface area of smaller prism) = 153 / 9
(surface area of smaller prism) = 17 cm²
Therefore, the surface area of the smaller prism is 17 cm².
Explanation: