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

Question

asked Apr 16, 2022 112k views

1 Answer

Ianjs · Answer 1 · 2022-04-20T08:01:06+0000

Answer:

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

Explanation:

Given hyperbola,

3x^2-y^2=23

Differentiate both side w.r.t. t

(d)/(dt) [3x^2-y^2]=(d)/(dt)[23]

$\Rightarrow (d)/(dt) [3x^2]-(d)/(dt)[y^2]=0$

$\Rightarrow 3*(d)/(dt) [x^2]-2y(dy)/(dt)=0$

$\Rightarrow 3* 2x(dx)/(dt)-2y(dy)/(dt)=0$

$\Rightarrow 6x(dx)/(dt)=2y(dy)/(dt)$

$\Rightarrow (dx)/(dt) =(2y)/(6x) *(dy)/(dt) =(y)/(3r) *(dy)/(dt)$

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

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

$\Rightarrow 48-23=y^2\\\Rightarrow y=√(25) = 5$

(dy)/(dt)=4

Now,
(dx)/(dt) =(y)/(3r) *(dy)/(dt)

=(5* 4)/(3* 4) =(5)/(3)

$\Rightarrow (dx)/(dt) =1.88\bar{8}=1.9$

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

Complete question is attached in below.