Question

details
no additional details were added for this assignment.
majeczka/shutterstock
which dimensions would produce a replica that is geometrically similar to the eiffel tower?
65 m by 24.98 m by 24.98 m
162.5 m by 249.8 m by 249.8 m
225 m by 24.90 m by 24.90 m
81.25 m by 41.63 m by 41.63 m

Explanation:

Step1: Recall similarity concept

For two - geometrically similar solids, the ratios of corresponding side lengths are equal. The Eiffel Tower has a certain set of proportional side - length relationships. We need to check the ratios of the given dimensions.
Let's assume the original ratios of the Eiffel Tower's dimensions are in a certain proportion. We can check the ratios of the given options' side - lengths.
For a three - dimensional object, if the ratios of the three side - lengths in one option are the same as the ratios of the side - lengths of the original object, it is geometrically similar.
Let's consider the ratios of the side - lengths in each option.
For the first option with dimensions \(65\ m\) by \(24.98\ m\) by \(24.98\ m\), the ratio of the first side to the second side is \(\frac{65}{24.98}\approx2.6\).
For the second option with dimensions \(162.5\ m\) by \(249.8\ m\) by \(249.8\ m\), the ratio of the first side to the second side is \(\frac{162.5}{249.8}\approx0.65\).
For the third option with dimensions \(225\ m\) by \(24.90\ m\) by \(24.90\ m\), the ratio of the first side to the second side is \(\frac{225}{24.90}\approx9.04\).
For the fourth option with dimensions \(81.25\ m\) by \(41.63\ m\) by \(41.63\ m\), the ratio of the first side to the second side is \(\frac{81.25}{41.63}\approx1.95\).
If we assume the Eiffel Tower has a non - square base and a certain height - to - base side ratio, we need to find an option where the ratios of the three dimensions are in proportion to the original.
Let's assume the Eiffel Tower has a height \(h\) and base side lengths \(a\) and \(b\) (where \(a = b\) for a square - like base at the bottom). We want \(\frac{h_1}{a_1}=\frac{h_2}{a_2}\) and \(\frac{h_1}{b_1}=\frac{h_2}{b_2}\) for the original (\(h_1,a_1,b_1\)) and the replica (\(h_2,a_2,b_2\)).
If we consider the Eiffel Tower as a three - dimensional object with a height and two equal base side lengths (approximate base shape), we note that for geometric similarity, the ratios of the corresponding sides must be equal.
Let's assume the Eiffel Tower has a height \(H\) and base side length \(B\). We want to find an option where \(\frac{H_1}{B_1}=\frac{H_2}{B_2}\).
For the option \(81.25\ m\) by \(41.63\ m\) by \(41.63\ m\), if we assume the first value is the height and the second and third are the base side lengths, the ratio of height to base side length is \(\frac{81.25}{41.63}\approx1.95\).
We can also check by cross - multiplying and comparing ratios.
Let the original Eiffel Tower have height \(h\) and base side \(b\). For a similar replica with height \(h'\) and base side \(b'\), we need \(\frac{h}{b}=\frac{h'}{b'}\).

Answer:

81.25 m by 41.63 m by 41.63 m