The volumes of two similar solids are 1331 cubic meters and 27 cubic meters. The surface area of the larger one is 242 square meters. What is the surface area of the smaller one?
the volumes of 2 similar solids are proportional to the cube of their corresponding sides
the surface ares of two similar solids are proportional to the square of their corresponding sides
so side1^3/side2^3 = 1331 / 27 = 11^3/3^3
side1/side2 = 11/3
area1/area2 = 11^2/3^2
242/area2 = 121/9
121area2 = 2178
area1 = 2178/121 = 18
To find the surface area of the smaller solid, we need to use the concept of similarity between the two solids.
The ratio between the volumes of two similar solids is equal to the cube of the ratio of their corresponding side lengths.
In this case, the ratio of the volumes is:
(1331 cubic meters) / (27 cubic meters) = 49
So, the ratio of the side lengths of the two solids is the cube root of 49:
∛49 ≈ 3
This means that the corresponding side lengths of the two solids are in a ratio of 3:1.
Next, we need to calculate the surface area of the larger solid, which is given as 242 square meters.
The surface area of a solid is directly proportional to the square of its side length. So, if the side length of the larger solid is 3x, then its surface area is:
(3x)^2 = 9x^2
And since the surface area is given as 242 square meters, we can set up the following equation:
9x^2 = 242
To solve for x, we divide both sides by 9:
x^2 = 242 / 9
x^2 ≈ 26.89
x ≈ √26.89
x ≈ 5.18
Now that we know x, we can find the surface area of the smaller solid, which is a 1:3 scale version of the larger solid.
If the side length of the larger solid is 3x, then the side length of the smaller solid is x.
The surface area of the smaller solid is then:
x^2 ≈ (5.18)^2
x^2 ≈ 26.77
Therefore, the surface area of the smaller solid is approximately 26.77 square meters.