The exact length of the curve given the following system of inequalities is ≈ 1637.
A system of inequalities refers to a set of two or more inequalities with one or more variables. This kind of system is used when a problem needs a range of solutions a there is over one constraint.
Step One - Let's restate the equations
We have:
x = 5 + 9t²
y = 4 + 6t³
Where
0 ≤ t ≤ 3
Step 2 - Differentiate them
The first derivative of dx/dt
= d/dt [9t² + 5)
= 9 * (d/dt) (t²) + (d/dt) (5)
= 9.2t + 0
= 18t
Also differentiate (dy/dt)
= d/dt [6t² + 4]
= 6 * (d/dt) [t³] + (d/dt) [4]
= 6.3 t² + 0
= 18t²
To find the length of the arc:
L = [tex]\[ \int_{0}^{4} \sqrt{(\frac{dx}{dt})^{2} + (\frac{dy}{dt})^{2} dt }[/tex]
We can thus deduce that:
= [tex]\[ \int_{0}^{4} \sqrt{(\fra18t)^{2} + ({18t^{2} )^{2} dt }[/tex]
= [tex]\int_{0}^{4}[18t \sqrt{1 + {18t^{2} ][/tex]
Compute the definite integral and factor out the constraints and we have:
dt = 4912/3
≈ 1,637.3
Hence the exact length of the curve is
≈ 1637
Learn more about the system of inequalities at:
https://brainly.com/question/9774970
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