`t=sqrtx+4` then find `((dx)/(dt))_(t=4)`

Options

A. 0
B. 1
C. 4
D. -1

Answer

To solve the problem, we start with the given equation:

\[ t = \sqrt{x} + 4 \]

We need to find \(\frac{dx}{dt}\) when \(t = 4\).

### Step 1: Rearranging the equation

First, we can rearrange the equation to express \(\sqrt{x}\) in terms of \(t\):

\[ \sqrt{x} = t - 4 \]

### Step 2: Squaring both sides

Next, we square both sides to solve for \(x\):

\[ x = (t - 4)^2 \]

### Step 3: Differentiating with respect to \(t\)

Now, we differentiate \(x\) with respect to \(t\):

\[ \frac{dx}{dt} = \frac{d}{dt}((t - 4)^2) \]

Using the chain rule, we find:

\[ \frac{dx}{dt} = 2(t - 4) \cdot \frac{d}{dt}(t - 4) = 2(t - 4) \cdot 1 = 2(t - 4) \]

### Step 4: Evaluating at \(t = 4\)

Now, we substitute \(t = 4\) into the derivative:

\[ \frac{dx}{dt} \bigg|_{t=4} = 2(4 - 4) = 2 \cdot 0 = 0 \]

### Conclusion

Thus, the value of \(\frac{dx}{dt}\) when \(t = 4\) is:

\[ \frac{dx}{dt} = 0 \]

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