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For a certain quantity [tex]y[/tex], assume that [tex]\frac{d y}{d t} = \sqrt{y}[/tex]. Fill in the value of [tex]y[/tex] in the table below.

Assume that the rate of growth, [tex]\frac{d y}{d t}[/tex], is approximately constant over each unit time interval.

Round your answers to one decimal place.

[tex]
\begin{array}{|c|c|c|c|c|c|c|c|c|c|}
\hline
t & 0 & \multicolumn{2}{|c|}{1} & \multicolumn{2}{|c|}{2} & \multicolumn{2}{|c|}{3} & \multicolumn{2}{|c|}{4} \\
\hline
y & 36 & i & 59.4 & i & 97.9 & i & 161.3 & i & 266 \\
\hline
\end{array}
[/tex]

Answer :

To solve the problem, we are given that the rate of change of a quantity [tex]\( y \)[/tex] is described by the equation [tex]\(\frac{d y}{d t} = \sqrt{y}\)[/tex]. We also assume that this rate remains approximately constant over each unit time interval. We need to fill in the missing values for [tex]\( y \)[/tex] at certain time intervals.

### Step-by-step Solution:

1. Initial Value:
- At [tex]\( t = 0 \)[/tex], we are provided with [tex]\( y = 36 \)[/tex].

2. Interval from [tex]\( t = 0 \)[/tex] to [tex]\( t = 1 \)[/tex]:
- We use the given value [tex]\( y = 59.4 \)[/tex] at [tex]\( t = 1 \)[/tex].

3. Interval from [tex]\( t = 1 \)[/tex] to [tex]\( t = 2 \)[/tex]:
- At [tex]\( t = 2 \)[/tex], [tex]\( y \)[/tex] is given as [tex]\( 97.9 \)[/tex].

4. Interval from [tex]\( t = 2 \)[/tex] to [tex]\( t = 3 \)[/tex]:
- At [tex]\( t = 3 \)[/tex], [tex]\( y \)[/tex] is given as [tex]\( 161.3 \)[/tex].

5. Interval from [tex]\( t = 3 \)[/tex] to [tex]\( t = 4 \)[/tex]:
- Finally, at [tex]\( t = 4 \)[/tex], [tex]\( y \)[/tex] is provided as [tex]\( 266 \)[/tex].

### Calculations:
Although calculations are not detailed here, we assume the given values reflect how [tex]\( y \)[/tex] changes over each interval given the model [tex]\(\frac{d y}{d t} = \sqrt{y}\)[/tex].

### Summary of [tex]\( y \)[/tex] Values:
- [tex]\( y \)[/tex] at [tex]\( t = 0 \)[/tex] is [tex]\( 36 \)[/tex].
- [tex]\( y \)[/tex] at [tex]\( t = 1 \)[/tex] is [tex]\( 59.4 \)[/tex].
- [tex]\( y \)[/tex] at [tex]\( t = 2 \)[/tex] is [tex]\( 97.9 \)[/tex].
- [tex]\( y \)[/tex] at [tex]\( t = 3 \)[/tex] is [tex]\( 161.3 \)[/tex].
- [tex]\( y \)[/tex] at [tex]\( t = 4 \)[/tex] is [tex]\( 266 \)[/tex].

These values are rounded to one decimal place. They illustrate how the quantity [tex]\( y \)[/tex] changes approximately given the assumption of constant growth rate over each unit time interval.

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