Question

If \(t = \sqrt{x} + 4\), then \(\left.\frac{dx}{dt}\right|_{t=4}\) is

• A. 4
• B. 0
• C. 8
• D. 16

Answer 1

The Correct Option is B

To solve this problem, we need to find the value of \(\left.\frac{dx}{dt}\right|_{t=4}\), given \(t = \sqrt{x} + 4\).

1. Start by expressing \(x\) in terms of \(t\). From the equation \(t = \sqrt{x} + 4\), we can solve for \(\sqrt{x}\) as follows: t - 4 = \sqrt{x}.
2. Now, square both sides to express \(x\) in terms of \(t\): x = (t - 4)^2.
3. Differentiate \(x = (t - 4)^2\) with respect to \(t\) to find \(\frac{dx}{dt}\):
   \frac{dx}{dt} = \frac{d}{dt}((t - 4)^2) = 2(t - 4).
4. Evaluate \(\frac{dx}{dt}\) at \(t = 4\): \left.\frac{dx}{dt}\right|_{t=4} = 2(4 - 4) = 2 \times 0 = 0.

The correct answer is 0, since at \(t = 4\), the derivative \(\frac{dx}{dt}\) becomes zero.