Answer :
To solve this problem, we need to understand the effect of dilation on the surface area of the bagel because the amount of cream cheese required is directly related to the surface area of the bagel.
Let's break it down step-by-step:
1. Understanding Dilation:
When an object undergoes dilation, its linear dimensions (length, width, height) change by a dilation factor. If we denote the dilation factor by [tex]\( d \)[/tex], each linear dimension of the bagel is increased by this factor [tex]\( d \)[/tex].
2. Surface Area Calculation:
Since cream cheese is applied as a thin layer on the surface of the bagel, we are interested in how the surface area changes with dilation. For a 3-dimensional object, the surface area increases by the square of the dilation factor.
[tex]\[ \text{New surface area} = (\text{original surface area}) \times d^2 \][/tex]
3. Comparing Surface Areas:
To find out how many times more cream cheese will be required for the dilated bagel, we compare the new surface area with the original surface area.
[tex]\[ \text{Ratio of cream cheese required} = d^2 \][/tex]
For instance, if the dilation factor [tex]\( d \)[/tex] is 2, the ratio of the surface area (and thus the cream cheese required) would be:
[tex]\[ \text{Ratio} = 2^2 = 4 \][/tex]
Thus, the dilated bagel would require 4 times more cream cheese than the original.
Since the problem asks for rounding to the nearest tenth and does not specify the dilation factor [tex]\( d \)[/tex], we should default to the example provided (commonly [tex]\( d = 2 \)[/tex]):
[tex]\[ \text{Answer} = 4.0 \][/tex]
So, the amount of cream cheese required for the dilated bagel would be 4.0 times more than for the original.
Let's break it down step-by-step:
1. Understanding Dilation:
When an object undergoes dilation, its linear dimensions (length, width, height) change by a dilation factor. If we denote the dilation factor by [tex]\( d \)[/tex], each linear dimension of the bagel is increased by this factor [tex]\( d \)[/tex].
2. Surface Area Calculation:
Since cream cheese is applied as a thin layer on the surface of the bagel, we are interested in how the surface area changes with dilation. For a 3-dimensional object, the surface area increases by the square of the dilation factor.
[tex]\[ \text{New surface area} = (\text{original surface area}) \times d^2 \][/tex]
3. Comparing Surface Areas:
To find out how many times more cream cheese will be required for the dilated bagel, we compare the new surface area with the original surface area.
[tex]\[ \text{Ratio of cream cheese required} = d^2 \][/tex]
For instance, if the dilation factor [tex]\( d \)[/tex] is 2, the ratio of the surface area (and thus the cream cheese required) would be:
[tex]\[ \text{Ratio} = 2^2 = 4 \][/tex]
Thus, the dilated bagel would require 4 times more cream cheese than the original.
Since the problem asks for rounding to the nearest tenth and does not specify the dilation factor [tex]\( d \)[/tex], we should default to the example provided (commonly [tex]\( d = 2 \)[/tex]):
[tex]\[ \text{Answer} = 4.0 \][/tex]
So, the amount of cream cheese required for the dilated bagel would be 4.0 times more than for the original.
