Answer:
[tex]176.4 \, \textsf{m}^2[/tex]
Step-by-step explanation:
If two cuboids are similar, the ratio of their surface areas is equal to the square of the ratio of their corresponding lengths.
Let the length of the yellow cuboid be [tex]L_{\textsf{yellow}} = 5[/tex] and
its surface area be [tex]A_{\textsf{yellow}} = 90 \, \textsf{m}^2[/tex].
Let the length of the red cuboid be [tex]L_{\textsf{red}} = 7[/tex] m, and we need to find its surface area [tex]A_{\textsf{red}}[/tex].
The ratio of their lengths is [tex] \dfrac{L_{\textsf{red}}}{L_{\textsf{yellow}}} = \dfrac{7}{5} [/tex].
The ratio of their surface areas is:
[tex] \left(\dfrac{L_{\textsf{red}}}{L_{\textsf{yellow}}}\right)^2 = \left(\dfrac{7}{5}\right)^2\\\\ = \dfrac{49}{25} [/tex].
Now, we can set up the proportion:
[tex]\dfrac{A_{\textsf{red}}}{A_{\textsf{yellow}}} = \dfrac{49}{25}[/tex]
We can solve for [tex]A_{\textsf{red}}[/tex]:
[tex]A_{\textsf{red}} = A_{\textsf{yellow}} \times \dfrac{49}{25} \\\\= 90 \, \textsf{m}^2 \times \dfrac{49}{25} \\\\ \approx 176.4 \, \textsf{m}^2[/tex]
Therefore, the surface area of the larger (red) cuboid is approximately [tex]176.4 \, \textsf{m}^2[/tex].
