41.8k views
5 votes
A solid has a volume of 128 cubic units and a surface area of 160 square units. The solid is dilated, and the image has a volume of 54 cubic units.

What is the surface area of the image?

User Sayuri
by
7.7k points

1 Answer

4 votes

Answer:

90 units²

Explanation:

The scale factor of the lengths of the sides of a solid must be squared to to find the ratio of the areas and cubed to find the ratio of the volumes.

For example, if you are given two solids with a scale factor of the lengths of the sides of 2, meaning that the sides of the second solid are twice the length of the sides of the first solid, then the surface area of the second solid is 2² (= 4) times the surface area of the original solid, and the volume of the second solid is 2² (= 8) times the volume of the original solid.

Let's say you have a cube with edge of 1 cm. The volume is (1 cm)³ = 1 cm³.

The surface area is the sum of the areas of the 6 faces which are squares 1 cm by 1 cm. 6 × (1 cm)² = 6 × 1 cm² = 6 cm².

The original cube with edge of 1 cm has a volume of 1 cm³ and total surface area of 6 cm².

Now double the size of the edge of the cube. The new cube has an edge of 2 cm. The volume is (2 cm)³ = 8 cm³. Each face of the cube is a square 2 cm by 2 cm. The total surface area of the new cube is 6 × (2 cm)² = 24 cm².

The new cube has volume 8 cm³ and total surface area 26 cm².

Compare the edges, surface areas and volumes of the two cubes:

Original Cube Dilated Cube Ratio new/dilated

Edge 1 cm 2 cm 1:2

Surface area 6 cm² 24 cm² 1:4

Volume 1 cm³ 8 cm³ 1:8

Now here is your problem.

We start with the ratio of the volumes.

Original solid: 128 u³

Dilated solid: 54 u³

ratio of volumes: 54/128 = 27/64 = 3³/4³

The ratio of volumes is the cube of the ratio of the lengths.

ratio of lengths: 3/4

ratio of area: 3²/4² = 9/16

Surface area of original solid: 160 u² × 9/16 = 90 u²

Answer: 90 units²

User Pouya Jabbarisani
by
6.9k points