the base of a right prism is a rhombus whose diagonals are 6 in and 8 in. The altitude of the prism measures 10 in. Find the lateral area and the total area of the prism.
To answer this question, we need to understand the concept of lateral area and total surface area of a prism.
Lateral area refers to the sum of the areas of all the lateral faces of a prism, while the total surface area includes the areas of both the lateral and base faces of the prism.
Given that the base of the right prism is a rhombus with diagonals measuring 6 inches and 8 inches, and the altitude (height) of the prism is 10 inches, we can apply the following formulas:
1. Lateral Area of a Prism:
The lateral area of a prism can be calculated by multiplying the perimeter of the base by the height of the prism.
2. Total Surface Area of a Prism:
The total surface area of a prism can be calculated by summing up the lateral area with the sum of the areas of both base faces.
Now, let's calculate the lateral area and total surface area of the given prism.
First, we need to find the perimeter of the base of the rhombus.
The perimeter of a rhombus can be calculated by adding four times the length of one side.
Since the diagonals of the rhombus are 6 inches and 8 inches, we can use the formula to find the length of one side:
Side = (1/2) * Square root of (diagonal1^2 + diagonal2^2)
Side = (1/2) * Square root of (6^2 + 8^2)
Side = (1/2) * Square root of (36 + 64)
Side = (1/2) * Square root of 100
Side = (1/2) * 10
Side = 5 inches
Now, we can calculate the perimeter of the base by multiplying the side length by 4:
Perimeter = 4 * Side
Perimeter = 4 * 5
Perimeter = 20 inches
For the lateral area, we multiply the perimeter of the base by the height of the prism:
Lateral Area = Perimeter * Height
Lateral Area = 20 inches * 10 inches
Lateral Area = 200 square inches
For the total surface area, we need to calculate the area of the base of the prism. Since the base is a rhombus, we can use the formula:
Area of a Rhombus = (diagonal1 * diagonal2) / 2
Area of the Base = (6 inches * 8 inches) / 2
Area of the Base = 48 square inches
Total Surface Area = Lateral Area + 2 * Area of the Base
Total Surface Area = 200 square inches + 2 * 48 square inches
Total Surface Area = 296 square inches
So, the lateral area of the prism is 200 square inches, and the total surface area of the prism is 296 square inches.