A rectangular Container $$ X $$ measured 85 cm long by 26 cm wide by 55 cm high. It was filled with apple juice to the brim. Another rectangular Container $$ Y $$, with a $$ 40-\mathrm{cm} $$ square base, was empty. The apple juice in Container $$ X $$ was poured into Container $$ Y $$ until the heights of the apple juice level in both containers were the same. Find the height of the apple juice level in both containers after the transfer. (Express the answer to 2 decimal places.)

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To find the height of the apple juice level in both containers after the transfer, we need to calculate the volume of the apple juice in Container $$ X $$ and then determine the height of the apple juice level in Container $$ Y $$ when the volumes are equal.

Given:
- Container $$ X $$ dimensions: 85 cm long, 26 cm wide, 55 cm high
- Container $$ Y $$ dimensions: 40 cm square base

Let's calculate the volume of the apple juice in Container $$ X $$:
$$ \text{Volume of apple juice in Container } X = \text{Length} \times \text{Width} \times \text{Height} $$

$$ \text{Volume of apple juice in Container } X = 85 \, \text{cm} \times 26 \, \text{cm} \times 55 \, \text{cm} $$

Now, let's calculate the volume of the apple juice in Container $$ X $$ and then determine the height of the apple juice level in Container $$ Y $$ when the volumes are equal.
Calculate the value by following steps:
- step0: Calculate:
  $$85\times 26\times 55$$
- step1: Multiply the terms:
  $$2210\times 55$$
- step2: Multiply the numbers:
  $$121550$$
The volume of the apple juice in Container $$ X $$ is 121550 cubic centimeters.

Now, let's calculate the height of the apple juice level in Container $$ Y $$ when the volumes are equal.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{121550}{\left(40\times 40\right)}$$
- step1: Remove the parentheses:
  $$\frac{121550}{40\times 40}$$
- step2: Multiply the numbers:
  $$\frac{121550}{1600}$$
- step3: Reduce the fraction:
  $$\frac{2431}{32}$$
The height of the apple juice level in both containers after the transfer is 75.97 cm (rounded to 2 decimal places).
The height of the apple juice level in both containers after the transfer is 75.97 cm.

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