Question

A point moves on the hyperbola so that its y-coordinate is increasing at a constant rate of 4 units per second. How fast is the x-coordinate changing when x

Answer 1

Answer:

x-coordinate is changing at a rate of 1.9 units per second.

Step-by-step explanation:

Given hyperbola,

  $3x^2-y^2=23$

Differentiate both side w.r.t. t

   $\frac{d}{dt} [3x^2-y^2]=\frac{d}{dt}[23]$

$\Rightarrow \frac{d}{dt} [3x^2]-\frac{d}{dt}[y^2]=0$

$\Rightarrow 3\times\frac{d}{dt} [x^2]-2y\frac{dy}{dt}=0$

$\Rightarrow 3\times 2x\frac{dx}{dt}-2y\frac{dy}{dt}=0$

$\Rightarrow 6x\frac{dx}{dt}=2y\frac{dy}{dt}$

$\Rightarrow \frac{dx}{dt} =\frac{2y}{6x} \times\frac{dy}{dt} =\frac{y}{3r} \times\frac{dy}{dt}$

$\Rightarrow 3x^2-y^2 = 23$

$\Rightarrow 3\times 4^2-y^2 =23 \quad \quad \text{when} \; x=4$

$\Rightarrow 48-23=y^2\\\Rightarrow y=\sqrt{25} = 5$

$\frac{dy}{dt}=4$

Now, $\frac{dx}{dt} =\frac{y}{3r} \times\frac{dy}{dt}$

             $=\frac{5\times 4}{3\times 4} =\frac{5}{3}$

    $\Rightarrow \frac{dx}{dt} =1.88\bar{8}=1.9$

Hence, the x-coordinate is changing at a rate of 1.9 units per second.

Complete question is attached in below.