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
Read more about rates of change at:
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