A particle is moving along the parabola x^2=4(y+6). As the particle passes through the point (8,10), the rate of change of its y-coordinate is 5 units per second. How fast, in units per second, is the x-coordinate changing at this instant?

Question

03-07-2017
Mathematics

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A particle is moving along the parabola x^2=4(y+6). As the particle passes through the point (8,10), the rate of change of its y-coordinate is 5 units per second. How fast, in units per second, is the x-coordinate changing at this instant?

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altavistard altavistard · Answer 1 · 2017-07-03T05:30:10+02:00

You'll need to use differentiation (specifically, implicit differentiation) here.

If x^2 = 4(y+6), differentiating both sides with respect to time t produces the following:

2x (dx/dt) = 4([dy/dt]) (note that (d/dt) 6 = 0)

We need to solve for (dx/dt). Substitute 8 for x (y does not appear in this latest equation, so we do nothing with y=10). Substitute the given 5 units/sec for dy/dt:

2(8)(dx/dt) = 4(5)(units/sec)

Solving for dx/dt, dx/dt = [20 units/sec]/16, or 5/4 units/sec, or 1.25 units/sec.

MrRoyal MrRoyal · Answer 2 · 2021-10-08T13:42:20+02:00

The instantaneous rate of change is the rate of change at a particular instant.

The rate of change per second in the x-coordinate is 1.25 units per second

The parabola is given as:

[tex]\mathbf{x^2 = 4(y + 6)}[/tex]

The rate of change of y is given as:

[tex]\mathbf{\frac{dy}{dt} = 5}[/tex]

Start by differentiating both sides, implicitly

[tex]\mathbf{2x\frac{dx}{dt} = 4\frac{dy}{dt}}[/tex]

Divide both sides by 2x

[tex]\mathbf{\frac{dx}{dt} = \frac{4dy}{2xdt}}[/tex]

From the question, we have:

[tex]\mathbf{(x,y) = (8,10)}[/tex]

Substitute 8 for x in [tex]\mathbf{\frac{dx}{dt} = \frac{4dy}{2xdt}}[/tex]

[tex]\mathbf{\frac{dx}{dt} = \frac{4dy}{2 \times 8dt}}[/tex]

Substitute [tex]\mathbf{\frac{dy}{dt} = 5}[/tex]

[tex]\mathbf{\frac{dx}{dt} = \frac{4 \times 5}{2 \times 8}}[/tex]

[tex]\mathbf{\frac{dx}{dt} = \frac{20}{16}}[/tex]

[tex]\mathbf{\frac{dx}{dt} = 1.25}[/tex]

Hence, the rate of change per second in the x-coordinate is 1.25 units per second