Multiple Choice 1 Point A Birthday Display Has A Large Gift Box In The Shape Of A Rectangular Prism With the Dimensions Shown Below. 10 Ft 8 Ft 5.5 Ft What Is The Total Surface Area Of The Box In Square Feet? F 198 Ft² G 440 Ft² H 259 Ft² J 358 Ft² 8.7B.1

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Understanding the Problem

In this problem, we are given the dimensions of a rectangular prism, which is a three-dimensional shape with length, width, and height. The dimensions of the prism are 10 ft, 8 ft, and 5.5 ft. We are asked to find the total surface area of the box in square feet.

What is Surface Area?

The surface area of a three-dimensional shape is the total area of all its faces. For a rectangular prism, the surface area can be calculated by finding the area of each face and adding them together.

Calculating the Surface Area of Each Face

To calculate the surface area of each face, we need to find the area of each individual face. The formula for the area of a rectangle is length x width.

  • Front Face: The front face has a length of 10 ft and a width of 8 ft. The area of the front face is 10 x 8 = 80 ft².
  • Back Face: The back face has the same dimensions as the front face, so its area is also 80 ft².
  • Top Face: The top face has a length of 10 ft and a width of 5.5 ft. The area of the top face is 10 x 5.5 = 55 ft².
  • Bottom Face: The bottom face has the same dimensions as the top face, so its area is also 55 ft².
  • Left Face: The left face has a length of 8 ft and a width of 5.5 ft. The area of the left face is 8 x 5.5 = 44 ft².
  • Right Face: The right face has the same dimensions as the left face, so its area is also 44 ft².
  • Left and Right Faces: Since the left and right faces are opposite each other, we only need to count them once. The total area of the left and right faces is 44 x 2 = 88 ft².
  • Front and Back Faces: Similarly, the front and back faces are opposite each other, so we only need to count them once. The total area of the front and back faces is 80 x 2 = 160 ft².
  • Top and Bottom Faces: Finally, the top and bottom faces are opposite each other, so we only need to count them once. The total area of the top and bottom faces is 55 x 2 = 110 ft².

Calculating the Total Surface Area

Now that we have the area of each face, we can add them together to find the total surface area of the box.

Total Surface Area = Area of Front Face + Area of Back Face + Area of Top Face + Area of Bottom Face + Area of Left Face + Area of Right Face + Area of Left and Right Faces + Area of Front and Back Faces + Area of Top and Bottom Faces Total Surface Area = 80 + 80 + 55 + 55 + 44 + 44 + 88 + 160 + 110 Total Surface Area = 658 ft²

However, we need to consider the faces that are shared by two adjacent faces. The left and right faces share a common edge with the front and back faces, and the top and bottom faces share a common edge with the left and right faces. To avoid counting these shared edges twice, we need to subtract the area of the shared edges from the total surface area.

The area of the shared edges is equal to the length of the shared edge multiplied by the width of the shared edge. The shared edges are 10 ft x 5.5 ft (top and bottom faces) and 8 ft x 5.5 ft (left and right faces).

Area of Shared Edges = (10 x 5.5) x 2 + (8 x 5.5) x 2 Area of Shared Edges = 110 + 88 Area of Shared Edges = 198 ft²

Now, we subtract the area of the shared edges from the total surface area to get the final answer.

Total Surface Area = 658 - 198 Total Surface Area = 460 ft²

However, this is not among the options provided. Let's re-evaluate our calculation.

We made an error in our calculation. The correct calculation is as follows:

Total Surface Area = 2 x (10 x 8) + 2 x (10 x 5.5) + 2 x (8 x 5.5) Total Surface Area = 160 + 110 + 88 Total Surface Area = 358 ft²

Q: What is the formula for calculating the total surface area of a rectangular prism?

A: The formula for calculating the total surface area of a rectangular prism is:

Total Surface Area = 2 x (length x width) + 2 x (length x height) + 2 x (width x height)

Q: How do I calculate the area of each face of a rectangular prism?

A: To calculate the area of each face of a rectangular prism, you need to multiply the length and width of each face. For example, if the length is 10 ft and the width is 8 ft, the area of the front face is 10 x 8 = 80 ft².

Q: What is the difference between the front face and the back face of a rectangular prism?

A: The front face and the back face of a rectangular prism are opposite each other and have the same dimensions. Therefore, they have the same area.

Q: How do I calculate the area of the shared edges of a rectangular prism?

A: To calculate the area of the shared edges of a rectangular prism, you need to multiply the length of the shared edge by the width of the shared edge. For example, if the length of the shared edge is 10 ft and the width of the shared edge is 5.5 ft, the area of the shared edge is 10 x 5.5 = 55 ft².

Q: Why do I need to subtract the area of the shared edges from the total surface area?

A: You need to subtract the area of the shared edges from the total surface area because the shared edges are counted twice when you calculate the total surface area. By subtracting the area of the shared edges, you avoid counting them twice and get the correct total surface area.

Q: What is the total surface area of a rectangular prism with dimensions 10 ft, 8 ft, and 5.5 ft?

A: The total surface area of a rectangular prism with dimensions 10 ft, 8 ft, and 5.5 ft is 358 ft².

Q: How do I apply the formula for calculating the total surface area of a rectangular prism to a real-world problem?

A: To apply the formula for calculating the total surface area of a rectangular prism to a real-world problem, you need to identify the length, width, and height of the prism and plug these values into the formula. For example, if you are designing a box with dimensions 10 ft, 8 ft, and 5.5 ft, you can use the formula to calculate the total surface area of the box.

Q: What are some common mistakes to avoid when calculating the total surface area of a rectangular prism?

A: Some common mistakes to avoid when calculating the total surface area of a rectangular prism include:

  • Not identifying the length, width, and height of the prism correctly
  • Not plugging the correct values into the formula
  • Not subtracting the area of the shared edges from the total surface area
  • Not double-checking the calculation for errors

By avoiding these common mistakes, you can ensure that your calculation of the total surface area of a rectangular prism is accurate and reliable.