Question

Find the length of the polar curve given by r=10e⁰.⁵θ on the interval 0≤θ≤1/6
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Answer 1

The length of the polar curve given by r = 10e^(0.5θ) on the interval 0 ≤ θ ≤ 1/6 is approximately 48.2 units.

Step-by-step explanation:

The length of a polar curve can be found using the following formula:

length = ∫√(r^2 + (dr/dθ)^2) dθ

In this case, we have r = 10e^(0.5θ).

Using this formula, we can find the length of the polar curve on the interval 0 ≤ θ ≤ 1/6:

length = ∫√(100e^θ + 25e^θ) dθ

= ∫√(125e^θ) dθ

= ∫√(5^3e^θ) dθ

Using the power rule of integration, we can find the antiderivative:

= 2/3 * (5^3/2 * e^(θ/2)) + C

= 2/3 * 125^(3/2) * e^(1/12) - 2/3 * 125^(3/2)

Now, we can evaluate the length on the given interval:

length = 2/3 * 125^(3/2) * e^(1/12) - 2/3 * 125^(3/2)

length ≈ 48.2 units.