Alex must wrap two boxes. Both boxes are right rectangular prisms. The smaller box has a length of 12 inches. The larger box is similar to the first and has a length of 48 inches. Find the ratio of the amount of paper needed to wrap the large box to the amount of paper needed to wrap the small box.
To find the ratio of the amount of paper needed to wrap the large box to the amount of paper needed to wrap the small box, we need to calculate the surface area of each box that is covered with paper when wrapping.
The surface area of a right rectangular prism can be found by adding up the areas of all its faces.
Let's start with the smaller box:
1. First, calculate the surface area of the smaller box. Since it is a right rectangular prism, it has 6 faces: 2 identical rectangular faces on the top and bottom, and 4 identical rectangular faces on the sides.
The rectangular faces on the top and bottom have dimensions of 12 inches (length) and 12 inches (width). So, the area of each of these faces is 12 inches * 12 inches = 144 square inches.
The rectangular faces on the sides have dimensions of 12 inches (width) and the depth of the box is not given. So, we need more information to calculate the areas of these faces.
Let's assume the depth of the smaller box is also 12 inches (since it is similar to the larger box). Therefore, the area of each side face is 12 inches * 12 inches = 144 square inches.
The total surface area of the smaller box is then: 2 * 144 square inches (top and bottom) + 4 * 144 square inches (sides) = 2 * 144 + 4 * 144 = 576 square inches.
Now, let's calculate the surface area of the larger box:
2. The larger box is similar to the smaller box, which means it has the same shape but is scaled up in size. Specifically, all corresponding dimensions are multiplied by the same constant factor.
In this case, the length of the larger box is 48 inches, which is 4 times the length of the smaller box. So, we can conclude that the larger box is a scaled-up version of the smaller box with a scaling factor of 4.
Since scaling by a factor of 4 multiplies the surface area by a factor of 4^2 = 16, the surface area of the larger box will be 16 times the surface area of the smaller box.
Therefore, the surface area of the larger box is: 16 * 576 square inches (surface area of the smaller box) = 9216 square inches.
Finally, we can find the ratio of the amount of paper needed to wrap the large box to the amount of paper needed to wrap the small box:
The ratio = Surface area of the larger box / Surface area of the smaller box
= 9216 square inches / 576 square inches
= 16
So, the ratio of the amount of paper needed to wrap the large box to the amount of paper needed to wrap the small box is 16:1.
To find the ratio of the amount of paper needed to wrap the large box to the amount of paper needed to wrap the small box, we need to consider the surface area of each box.
The surface area of a right rectangular prism can be calculated using the formula:
Surface Area = 2 * (length * width + length * height + width * height)
For the small box:
Length of small box = 12 inches
Assuming the width and height of the small box are unknown, we can represent them as variables (w and h respectively):
Surface Area of small box = 2 * (12w + 12h + wh)
For the large box:
Length of large box = 48 inches
Since the large box is similar to the small box, the ratio of the lengths is 48/12 = 4. This means that each dimension of the large box is 4 times the corresponding dimension of the small box.
Therefore, the width of the large box (W) = 4w
And the height of the large box (H) = 4h
Surface Area of large box = 2 * (48W + 48H + WH)
= 2 * (48 * 4w + 48 * 4h + 4w * 4h)
= 2 * (192w + 192h + 16wh)
= 384w + 384h + 32wh
Now, we can find the ratio of the surface areas:
Ratio = (Surface Area of large box) / (Surface Area of small box)
= (384w + 384h + 32wh) / (2 * (12w + 12h + wh))
Note: The exact value of the ratio cannot be determined without knowing the values of width and height for both boxes.