You are going to build a glass box. The length of the box is 8 inches more than the width and the height is half the length. The length of the wire to make the frame is 108 inches.
1.What are the dimensions of the box?
2.All six faces of the box will be panes of glass. What is the total area of the glass needed for the six faces?
3.What is the volume of the box?
width ---- x
length ---- x+8
height = x/2 + 4
framing consists of 4 lengths, 4 widths, and 4 heights (make a sketch to see)
4x + 4(x+8) + 4(x/2 + 4) = 108
4x + 4x + 32 + 2x + 16 = 108
10x = 60
x = 6
continue ....
1. To find the dimensions of the box, let's assign variables for the width, length, and height. Let's say:
- Width = W
- Length = W + 8 (since the length is 8 inches more than the width)
- Height = (W + 8) / 2 (since the height is half the length)
Now, we know the length of the wire used to make the frame is 108 inches. The wire goes around the perimeter of the box, so we can calculate the perimeter using the dimensions we identified above.
Perimeter = 2(Length + Width + Height)
108 = 2((W + 8) + W + (W + 8)/2)
108 = 2(2W + 8 + (W + 8)/2)
Simplifying the equation:
108 = 2(2W + 8 + (W + 8)/2)
108 = 4W + 16 + (W + 8)
108 = 5W + 24
5W = 108 - 24
5W = 84
W = 84/5
W = 16.8 inches
Now we can substitute the value of W back into the expressions for Length and Height to find the dimensions of the box:
Length = W + 8 = 16.8 + 8 = 24.8 inches
Height = (W + 8)/2 = (16.8 + 8)/2 = 24.8/2 = 12.4 inches
Therefore, the dimensions of the box are:
Width = 16.8 inches
Length = 24.8 inches
Height = 12.4 inches
2. To find the total area of glass needed for the six faces, we need to calculate the area of each face separately and then sum them up.
The box has 6 faces: 2 opposite faces with dimensions Length x Width, 2 opposite faces with dimensions Length x Height, and 2 opposite faces with dimensions Width x Height.
Area of each face can be calculated as: Length * Width
Area of the top and bottom faces = Length * Width = 24.8 * 16.8
Area of the front and back faces = Length * Height = 24.8 * 12.4
Area of the left and right faces = Width * Height = 16.8 * 12.4
Total area of the six faces = 2(Area of top and bottom faces) + 2(Area of front and back faces) + 2(Area of left and right faces)
Calculate the value using the dimensions we found in Step 1.
Therefore, the total area of glass needed for the six faces is:
2(24.8 * 16.8) + 2(24.8 * 12.4) + 2(16.8 * 12.4)
3. To find the volume of the box, we multiply the length, width, and height.
Volume = Length * Width * Height
Substituting the values we found in Step 1:
Volume = 24.8 * 16.8 * 12.4