Question

\(1^{2}=x^{2}+y^{2}\) \(1 \frac{d l}{d t}=x \frac{d x}{d t}+y \frac{d y}{d t}\) \(y=x^{3 / 2}\) \(\frac{d y}{d t}=\frac{3}{2} x^{1 / 2} \frac{d x}{d t}\) Using (1) \& (2) $11 \sqrt{x^{2}+x^{3}}=x \frac{d x}{d t}+\frac{3}{2} x^{3 / 2} x^{1 / 2} \frac{d x}{d t}$ \(\frac{d x}{d t}=\frac{66}{3+\frac{27}{2}}=\frac{66 \times 2}{33}=4\)

Short answer

In summary, the rate of change of \(x\) with respect to time, \(\frac{dx}{dt}\), in the given system of related rates, is equal to \(4\).

Step-by-step solution

Identify the given equations and variables

We have the following equations given to us: 1. \(1^2 = x^2 + y^2\) 2. \(1\frac{dl}{dt} = x\frac{dx}{dt} + y\frac{dy}{dt}\) 3. \(y = x^{\frac{3}{2}}\) 4. \(\frac{dy}{dt}=\frac{3}{2}x^{\frac{1}{2}}\frac{dx}{dt}\) Our objective is to find \(\frac{dx}{dt}\).

Use given equations to solve for y

From equation (3), we have: \(y = x^{\frac{3}{2}}\) Square both sides: \(y^2 = x^3\) Replace \(y^2\) in equation (1) using this result: \(1^2 = x^2 + x^3\) Simplify: \(1 = x^2 + x^3\)

Solve for \(\frac{dy}{dt}\)

From equation (4), we have: \(\frac{dy}{dt}=\frac{3}{2}x^{\frac{1}{2}}\frac{dx}{dt}\)

Replace \(y\) and \(\frac{dy}{dt}\) in equation (2)

Using equations (3) and (4), we can rewrite equation (2) as: \(1\frac{dl}{dt} = x\frac{dx}{dt} + x^{\frac{3}{2}}\left(\frac{3}{2}x^{\frac{1}{2}}\frac{dx}{dt}\right)\)

Solve for \(\frac{dx}{dt}\)

Now, we have a single equation that can be solved for \(\frac{dx}{dt}\): \(1\frac{dl}{dt} = x\frac{dx}{dt} + \frac{3}{2}x^2\frac{dx}{dt}\) Given: \(11\sqrt{x^2+x^3}=x\frac{dx}{dt}+\frac{3}{2}x^{\frac{3}{2}}\frac{dx}{dt}\) Multiply both sides by \(\frac{2}{3}\): \(\frac{22}{3}\sqrt{x^2+x^3}=\frac{2}{3}x\frac{dx}{dt}+\frac{1}{2}x^{\frac{3}{2}}\frac{dx}{dt}\) Combine the terms on right side: \(\frac{22}{3}\sqrt{x^2+x^3}=\left(\frac{2}{3}x+\frac{1}{2}x^{\frac{3}{2}}\right)\frac{dx}{dt}\) Now, divide both sides to find \(\frac{dx}{dt}\): \(\frac{dx}{dt}=\frac{\frac{22}{3}\sqrt{x^2+x^3}}{\frac{2}{3}x+\frac{1}{2}x^{\frac{3}{2}}}\) Given the provided input, \(\frac{dx}{dt}=\frac{66}{3+\frac{27}{2}}=\frac{66\times 2}{33}=4\) The rate of change of \(x\) with respect to time, \(\frac{dx}{dt}\), is equal to \(4\).