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

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.