Answer to Question #318515 in Geometry for shibs

Question

Find the distance, in inches between two vertices of a cube that are farthest 

From each other, if an edge measures 10 inches.

b) Compute the lateral surface area of the cube in square inches.

c) Compute the Total Surface area of the cube in inches squared

d) Compute the volume of the cube in cubic inches.

Answer 1

a) The distance between two vertices of a cube that are farthest from each other is the diagonal of a cube. For example, it can e the distance between vertices B and D₁.

AB = AD = DD₁ = 10 inches, because an edge measures 10 inches.

Let’s find DB using the Pythagorean theorem for the triangle DAB:

$DB=\sqrt{AB^2+AD^2} = \sqrt{10^2+10^2} = \sqrt{100+100} = \sqrt{200} = 10\sqrt{2}$ inches.

Let’s find BD₁ using the Pythagorean theorem for the triangle DBD₁:

$BD_1=\sqrt{DB^2+DD_1^2} = \sqrt{(10\sqrt2)^2+10^2} = \sqrt{200+100} = \sqrt{300}= 10\sqrt3 \approx 17.32$ inches.

ANSWER: The distance between the vertices farthest from each other in a cube is $10\sqrt3$  inches or approx. 17.32 inches.

b) Compute the lateral surface area of the cube in square inches.

The lateral surface area of a cube refers to the total area covered by the side or lateral faces of a cube. To calculate LSA, we find the sum of areas of these 4 faces where each is a square with the same sides. Area of each side is equal to a², where 'a' is the length of the edge of the cube. So the lateral surface area (LSA) is equal to:

LSA = 4a² = 4 * 10² = 4 * 100 = 400 sq. inches.

ANSWER: The lateral surface area of a cube is 400 sq. inches.

c) Compute the total surface area of the cube in square inches.

The total surface area of a cube refers to the total area covered by all six faces of a cube. To calculate the TSA of a cube, we find the sum of the areas of these 6 faces where each is a square with the same sides. Area of each side is equal to a², where 'a' is the length of the edge of the cube. So the total surface area (TSA) is equal to:

TSA = 6a² = 6 * 10² = 6 * 100 = 600 sq. inches.

ANSWER: The total surface area of a cube is 600 sq. inches.

d) Compute the volume of the cube in cubic inches.

The formula to calculate the volume (V) of a cube is given as, Volume = a³, where 'a' is the length of the side of the cube. S the volume of a cube is equal to:

V = a³ = 10³ = 1000 cubic inches.

ANSWER: The volume of the cube is 1000 cubic inches.