Find the arc length function for the curve $y=10 x^{3 / 2}$ with starting point $P_{0}(1,15)$ .
A) $L=\frac{2}{675}\left((1+225 x)^{3 / 2}+226\right)$
B) $L=\frac{2}{225}\left((1+220 x)^{3 / 2}-226 \sqrt{226}\right)$
C) $L=\frac{2}{675}\left((1+225 x)^{5 / 2}+226 \sqrt{226}\right)$
D) $L=\frac{2}{675}\left((1+225 x)^{3 / 2}-226 \sqrt{226}\right)$
E) $L=\frac{2}{675}\left((1+225 x)^{1 / 2}-226 \sqrt{226}\right)$

Question

Quizplus · Accepted Answer

The arc length function is derived by integrating the square root of $1 + (dy/dx)^2$. For $y = 10x^{3/2}$, $dy/dx = 15x^{1/2}$. The arc length integral becomes $\int \sqrt{1 + (15x^{1/2})^2} \, dx = \int \sqrt{1 + 225x} \, dx$. Solving this integral gives the arc length function, which matches option D.

Find the Arc Length Function for the Curve y=10x3/2y=10 x^{3 / 2}y=10x3/2

Find the Arc Length Function for the Curve $y=10 x^{3 / 2}$