exercises 3.8 implicit differentiation\nscore: 14/20 answered: 7/10\nprogress saved done\nquestion…

exercises 3.8 implicit differentiation\nscore: 14/20 answered: 7/10\nprogress saved done\nquestion 8\ndetails\nthe number of cars produced when x dollars is spent on labor and y dollars is spent on capital invested by a manufacturer can be modeled by the equation\n50x^{\\frac{1}{3}}y^{\\frac{2}{3}} = 600.\na. find a formula in terms of x and y for \\frac{dy}{dx}.\n\\frac{dy}{dx}=\n\nb. find the value of of \\frac{dy}{dx} at the point (27,8).\nround to 4 decimal places.\nquestion help: video message instructor\nsubmit question jump to answer

exercises 3.8 implicit differentiation\nscore: 14/20 answered: 7/10\nprogress saved done\nquestion 8\ndetails\nthe number of cars produced when x dollars is spent on labor and y dollars is spent on capital invested by a manufacturer can be modeled by the equation\n50x^{\\frac{1}{3}}y^{\\frac{2}{3}} = 600.\na. find a formula in terms of x and y for \\frac{dy}{dx}.\n\\frac{dy}{dx}=\n\nb. find the value of of \\frac{dy}{dx} at the point (27,8).\nround to 4 decimal places.\nquestion help: video message instructor\nsubmit question jump to answer

Answer

Explanation:

Step1: Differentiate both sides of the equation

Differentiate (50x^{\frac{1}{3}}y^{\frac{2}{3}} = 600) with respect to (x) using the product rule ((uv)^\prime=u^\prime v + uv^\prime), where (u = 50x^{\frac{1}{3}}) and (v = y^{\frac{2}{3}}).

The derivative of (u = 50x^{\frac{1}{3}}) with respect to (x) is (u^\prime=\frac{50}{3}x^{-\frac{2}{3}}).

Using the chain - rule, the derivative of (v = y^{\frac{2}{3}}) with respect to (x) is (v^\prime=\frac{2}{3}y^{-\frac{1}{3}}\frac{dy}{dx}).

Differentiating the left - hand side: (\frac{d}{dx}(50x^{\frac{1}{3}}y^{\frac{2}{3}})=50\left(\frac{1}{3}x^{-\frac{2}{3}}y^{\frac{2}{3}}+x^{\frac{1}{3}}\cdot\frac{2}{3}y^{-\frac{1}{3}}\frac{dy}{dx}\right))

The derivative of the right - hand side (\frac{d}{dx}(600) = 0).

So, (50\left(\frac{1}{3}x^{-\frac{2}{3}}y^{\frac{2}{3}}+\frac{2}{3}x^{\frac{1}{3}}y^{-\frac{1}{3}}\frac{dy}{dx}\right)=0)

Step2: Solve for (\frac{dy}{dx})

First, divide both sides of the equation (50\left(\frac{1}{3}x^{-\frac{2}{3}}y^{\frac{2}{3}}+\frac{2}{3}x^{\frac{1}{3}}y^{-\frac{1}{3}}\frac{dy}{dx}\right)=0) by (50) (since (50\neq0)):

(\frac{1}{3}x^{-\frac{2}{3}}y^{\frac{2}{3}}+\frac{2}{3}x^{\frac{1}{3}}y^{-\frac{1}{3}}\frac{dy}{dx}=0)

Subtract (\frac{1}{3}x^{-\frac{2}{3}}y^{\frac{2}{3}}) from both sides:

(\frac{2}{3}x^{\frac{1}{3}}y^{-\frac{1}{3}}\frac{dy}{dx}=-\frac{1}{3}x^{-\frac{2}{3}}y^{\frac{2}{3}})

Multiply both sides by (\frac{3}{2x^{\frac{1}{3}}y^{-\frac{1}{3}}})

(\frac{dy}{dx}=-\frac{y}{2x})

Step3: Evaluate (\frac{dy}{dx}) at the point ((x = 27,y = 8))

Substitute (x = 27) and (y = 8) into (\frac{dy}{dx}=-\frac{y}{2x})

(\frac{dy}{dx}\mid_{(27,8)}=-\frac{8}{2\times27}=-\frac{4}{27}\approx - 0.1481)

Answer:

a. (\frac{dy}{dx}=-\frac{y}{2x}) b. (-0.1481)