A triangular prism has a height of 1.5 inches and right triangular bases with a height of 1.5 inches, length of 5 inches, and a hypotenuse measuring 5.22 inches. What is the prism's surface area?
A: 25.08 in.2
B: 24.75 in.2
C: 25.41 in.2
D: 30.33 in.2
A triangular prism has a height of 1.5 inches and right triangular bases with a height of 1.5 inches, length of 5 inches, and a hypotenuse measuring 5.22 inches. What is the prism's surface area?
A: 25.08 inches squared.
B: 24.75 inches squared.
C: 25.41 inches squared.
D: 30.33 inches squared.
The area of each of the triangular bases is $$\frac{1}{2}(\text{base})(\text{height})=\frac{1}{2}(5)(1.5)=3.75.$$The three lateral faces each have an area of $$\frac{1}{2}(\text{base})(\text{height})=\frac{1}{2}(5.22)(1.5)=3.915.$$Therefore, the total surface area is $$3\cdot 3.915+2\cdot 3.75=\boxed{\textbf{(A) }25.08 \text{ in.}^2}.$$
i cant read that. can you write just in english?
Sure! Here's a simplified version of the solution:
The triangular prism has two triangular bases and three rectangular lateral faces. The area of each base is 3.75 square inches, and the area of each lateral face is 3.915 square inches. To find the total surface area, we can add up the areas of all five faces. This gives us:
Total surface area = (area of base) x 2 + (area of lateral face) x 3
Total surface area = 3.75 x 2 + 3.915 x 3
Total surface area = 25.08 square inches
Therefore, the surface area of the triangular prism is 25.08 square inches, which is answer choice (A).
To find the surface area of a triangular prism, we need to calculate the area of the two triangular bases and the three rectangular sides.
1. Calculate the area of the triangular bases:
a. The base of the triangular base is the length of one side of the right triangle, which is 5 inches.
b. The height of the triangular base is given as 1.5 inches.
c. Using the formula for the area of a triangle: A = 0.5 * base * height.
Substituting the values: A = 0.5 * 5 * 1.5 = 3.75 in.²
2. Calculate the area of the rectangular sides:
a. The length of the rectangular side is the same as the base of the right triangle, which is 5 inches.
b. The height of the rectangular side is equal to the height of the prism, given as 1.5 inches.
c. Using the formula for the area of a rectangle: A = length * height.
Substituting the values: A = 5 * 1.5 = 7.5 in.²
3. Now, we sum up the areas of both triangular bases and the three rectangular sides:
3.75 in.² (triangular base) + 3.75 in.² (triangular base) + 7.5 in.² (rectangular side) + 7.5 in.² (rectangular side) + 7.5 in.² (rectangular side) = 30 in.²
Based on the calculations, the surface area of the triangular prism is 30 in.², which is not provided as an option. Therefore, none of the answer choices A, B, C, or D is correct.
To find the surface area of a triangular prism, we need to calculate the areas of all the faces and then sum them.
A triangular prism has 3 pairs of congruent faces. Each pair consists of a rectangular face and a triangular face.
First, let's calculate the area of the rectangular faces.
The rectangular faces have dimensions of 1.5 inches (height) by 5 inches (length). To find the area, we multiply the height by the length: 1.5 * 5 = 7.5 square inches.
Since there are two rectangular faces, the total area of the rectangular faces is 2 * 7.5 = 15 square inches.
Next, let's calculate the area of the triangular faces.
The triangular faces are right triangles with one leg measuring 1.5 inches (height) and the hypotenuse measuring 5.22 inches.
To find the length of the other leg of the right triangle, we can use the Pythagorean theorem: a^2 + b^2 = c^2, where a and b are the legs and c is the hypotenuse.
In this case, we have 1.5^2 + b^2 = 5.22^2.
Simplifying, we get 2.25 + b^2 = 27.2484.
Subtracting 2.25 from both sides, we have b^2 = 24.9984.
Taking the square root of both sides, we get b ≈ 4.999 inches.
Now that we have the length of the other leg, the area of each triangular face is (1.5 * 4.999) / 2 = 3.74925 square inches.
Since there are two triangular faces, the total area of the triangular faces is 2 * 3.74925 = 7.4985 square inches.
Finally, to find the surface area of the triangular prism, we add the area of the rectangular faces and the area of the triangular faces: 15 + 7.4985 = 22.4985 square inches.
Rounded to two decimal places, we get 22.50 in.2.
Based on the given answer choices, the closest option is B: 24.75 in.2.