Question

Find the exact length of the curve. x = 5 + 9t2, y = 4 + 6t3, 0 ≤ t ≤ 3

Answer 1

The exact length of the curve given the following system of inequalities is ≈ 1637.

What is a system of inequalities?

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.

What is the length of the curve with the above system of inequalities?

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 = $\[ \int_{0}^{4} \sqrt{(\frac{dx}{dt})^{2} + (\frac{dy}{dt})^{2} dt }$

We can thus deduce that:

= $\[ \int_{0}^{4} \sqrt{(\fra18t)^{2} + ({18t^{2} )^{2} dt }$

= $\int_{0}^{4}[18t \sqrt{1 + {18t^{2} ]$

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:
brainly.com/question/9774970
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