Jennifer got a box of chocolates. The box is a right triangular prism shaped box. It is 7 inches long, and the triangular base measures 3in x 4in x 5in. What is the surface area of the box of chocolates?
A 3-4-5 triangle is a right triangle, so its area is 1/2 * 3*4 = 6
The sides are all rectangles of height 7.
So now just add up the areas of the two bases and the three sides.
To find the surface area of the right triangular prism-shaped box, we need to calculate the areas of all its faces and then add them together.
The box consists of three rectangular faces and two triangular faces. Let's begin with the rectangular faces.
The front and back face:
- The length of the box (7 inches) corresponds to the height of these rectangular faces.
- The width of the box will be the width of the triangular base, which is 3 inches.
- To find the area of one rectangular face, multiply the length and width.
Area of one rectangular face = length × width = 7 inches × 3 inches.
Since there are two identical rectangular faces, we multiply the area of one face by 2.
Total area of the front and back faces = 2 × (7 inches × 3 inches).
The top and bottom face:
- The length of the box (7 inches) corresponds to the length of these rectangular faces.
- The width of the box will be the length of the triangular base, which is 4 inches.
- To find the area of one rectangular face, multiply the length and width.
Area of one rectangular face = length × width = 7 inches × 4 inches.
Since there are two identical rectangular faces, we multiply the area of one face by 2.
Total area of the top and bottom faces = 2 × (7 inches × 4 inches).
Now let's move to the triangular faces.
The two triangular faces have the same dimensions as the base of the box, which is a right triangle with side lengths 3 inches, 4 inches, and 5 inches.
To find the area of a triangle, we can use Heron's formula. However, in this case, we can use a simpler approach since it's a right triangle.
The area of a right triangle = (base × height) / 2.
Area of one triangular face = (3 inches × 4 inches) / 2.
Since there are two identical triangular faces, we multiply the area of one face by 2.
Total area of the two triangular faces = 2 × [(3 inches × 4 inches) / 2].
Now we can sum up all the areas to find the total surface area of the box.
Total surface area = Total area of the front and back faces + Total area of the top and bottom faces + Total area of the two triangular faces.
Total surface area = [2 × (7 inches × 3 inches)] + [2 × (7 inches × 4 inches)] + [2 × [(3 inches × 4 inches) / 2]].
Calculating the values:
Total surface area = [2 × (21 inches²)] + [2 × (28 inches²)] + [2 × (6 inches²)].
Total surface area = 42 inches² + 56 inches² + 12 inches².
Total surface area = 110 inches².
The surface area of the box of chocolates is 110 square inches.