The sides and hypotenuse of a right triangle are strictly increasing with time. At the instant when x is 24 inches and y is 32 inches, dy/dt = 2 dx/dt. If dθ/dt = −0.01 radians per minute at the same instant, what is the value of dy/dt at that same instant ?

Answer:Step-by-step explanation:Did you perhaps mean what is the value of dx/dt at that instant?  You have a value for dy/dt to be 2dx/dt. I'm going with that, so if it is an incorrect assumption I have made, I apologize!Here's what we have:We have a right triangle with a reference angle (unknown as of right now), side y and side x; we also have values for y and x, and the fact that dθ/dt=-.01So the game plan here is to use the inverse tangent formula to solve for the missing angle, and then take the derivative of it to solve for dx/dt.Here's the inverse tangent formula:[tex]tan\theta=\frac{y}{x}[/tex]and its derivative:[tex]sec^2\theta\frac{d\theta }{dt} =\frac{x\frac{dy}{dt}-y\frac{dx}{dt}  }{x^2}}[/tex]We have values for y, x, dy/dt, and dθ/dt.  We only have to find the missing angle theta and solve for dx/dt.Solving for the missing angle first:[tex]tan\theta =\frac{32}{24}[/tex]On your calculator you will find that the inverse tangent of that ratio gives you an angle of 53.1°.Filling in the derivative formula with everything we have:[tex]sec^2(53.1)(-.01)=\frac{24\frac{dx}{dt}-32\frac{dx}{dt}  }{24^2}[/tex]We can simplify the left side down a bit by breaking up that secant squared like this:[tex]sec(53.1)sec(53.1)(-.01)[/tex]We know that the secant is the same as 1/cos, so we can make that substitution:[tex]\frac{1}{cos53.1} *\frac{1}{cos53.1} *-.01[/tex] and[tex]\frac{1}{cos53.1}=1.665500191[/tex]We can square that and then multiply in the -.01 so that the left side looks like this now, along with some simplification to the right:[tex]-.0277389=\frac{48\frac{dx}{dt} -32\frac{dx}{dt} }{576}[/tex]We will muliply both sides by 576 to get:[tex]-15.9776=48\frac{dx}{dt}-32\frac{dx}{dt}[/tex]We can now factor out the dx/dt to get:[tex]-15.9776=16\frac{dx}{dt}[/tex] (16 is the result of subtracting 32 from 48)Now we divide both sides by 16 to get that[tex]\frac{dx}{dt}=-.9986\frac{radians}{minute}[/tex]The negative sign obviously means that x is decreasing