The base of a right prism 50 inches high is a rhombus, with diagonals 30 and 16 inches. Find the total surface area of the prism.
To find the total surface area of the prism, we need to calculate the areas of all the individual faces and then add them together.
Let's start by finding the area of the base. The base of the prism is a rhombus, so we can use the formula:
Area of a rhombus = (diagonal1 * diagonal2) / 2
In this case, the diagonals are 30 inches and 16 inches. Plugging these values into the formula, we get:
Area of the base = (30 * 16) / 2 = 480 square inches
Since the prism has two bases, the total area of the base is 2 * 480 = 960 square inches.
Now let's find the lateral surface area of the prism. The lateral surface area is the sum of the areas of the four rectangular faces.
To find the area of each rectangular face, we need to know the height and the length of the sides. In this case, the height of the prism is given as 50 inches.
Looking at the rhombus base, we can see that the length of its sides corresponds to the height of the prism. Therefore, each side of the rhombus is 50 inches long.
To find the length of the sides of the rectangular faces, we can use the Pythagorean theorem. Since the diagonals of the rhombus bisect each other at right angles, the sides of the rhombus form right triangles.
Let's call the length of the side of the rhombus "a". Using the Pythagorean theorem, we can set up the following equation:
(a/2)^2 + (a/2)^2 = 16^2/2
Simplifying this equation, we get:
(a^2)/4 + (a^2)/4 = 256/2
Combining like terms, we have:
(a^2)/2 = 128
Multiplying both sides by 2, we get:
a^2 = 256
Taking the square root of both sides, we find:
a = 16
So, each side of the rhombus, and therefore each side of the rectangular faces, is 16 inches long.
Now we can find the area of one rectangular face:
Area of a rectangular face = length * height
In this case, the length is 16 inches and the height is 50 inches. Calculating the area, we get:
Area of each rectangular face = 16 * 50 = 800 square inches
Since there are four rectangular faces, the total lateral surface area is 4 * 800 = 3200 square inches.
Finally, we can find the total surface area of the prism by adding the area of the bases to the lateral surface area:
Total surface area = area of the bases + lateral surface area
= 960 + 3200
= 4160 square inches
Therefore, the total surface area of the prism is 4160 square inches.