Question

) The density of oil in a circular oil slick on the surface of the ocean at a distance of r meters from the center of the slick is given by δ(r)=401+r2 kilograms per square meter. Find the exact value of the mass of the oil slick if the slick extends from r=0 to r=5 meters.

Answer 1

Answer:

Therefore the mass of the of the oil is 409.59 kg.

Step-by-step explanation:

Let us consider a circular disk. The inner radius of the disk be r and the outer diameter of the disk be (r+Δr).

The area of the disk

=The area of the outer circle - The area of the inner circle

= $\pi (r+\triangle r)^2- \pi r^2$

$=\pi [r^2+2r\triangle r+(\triangle r)^2-r^2]$

$=\pi [2r\triangle r+(\triangle r)^2]$

Since (Δr)² is very small, So it is ignorable.

∴$A=2\pi r\triangle r$

The density $\delta (r)= \frac{40}{1+r^2}$

We know,

Mass= Area× density

        $=(2r \pi \triangle r)(\frac{40}{1+r^2}})$

Total mass $M=\sum_{i=1}^n \frac{80r_i\pi }{1+r^2}\triangle r_i$

Therefore

$\sum_{i=1}^n \frac{80r_i\pi }{1+r^2}\triangle r_i=\int_0^5 \frac{80r\pi }{1+r^2}dr$

                      $=40\pi[ln(1+r^2)]_0^5$

                      $=40\pi [ln(1+5^2)-ln(1+0^2)]$

                     $=40\pi ln(26)$

                     = 409.59 kg (approx)

Therefore the mass of the of the oil is 409.59 kg.