A car’s rear windshield wiper rotates 125^(circ) The total length of the wiper mechanism is 25 inches and wipes the windshield over a distance of 14 inches. Find the area covered by the wiper.

Three pipes, each of radius length 4 in., are stacked as shown. What is the exact height of the stack?

If the atomic radius of lead is 0.175 nm, calculate the volume of its unit cell in cubic meters

Two circles, whose radii are 12 inches and 16 inches respectively, intersect. The angle between the tangents at either of the points of intersection is ${\displaystyle {20}^{\circ },{30}^{\prime }}$. Find the distance between the centers of the circle.

Five circles are placed in a rectangle as shown. If the length of the shorter side of the rectangle is 1, find the length of the other side.

I am trying to compute this integral ${\int }_0^{\mathrm{\infty }}\frac{x}{1+x^6}dx$ using the residue theorem. To do so, I am integrating $f(z)=\frac{z}{1+z^6}$ in the frontier of this sector of a circle: $\{z:|z|<R,0<arg(z)<\pi /3\}$.

I kwo how to deal with the integrals over the horizontal segment of the sector and over the arc of the circle. My problem is the "diagonal segment". When I parametrize it, I do not get something easy to integrate. How could I approach this?

(The path of integration was suggested in my book, so I do not think that is the problem).

Suppose I'm given two points: $(x_1,y_1)$ and $(x_2,y_2)$ (which are real numbers) lying on the circumference of a circle with radius r and centred at the origin, how do I find the arc length between those two points (the arc with shorter length)?

The square surface cover shown in the figure is 10.16 centimeters on each side. The knockout in the center of the cover has a diameter of 1.27 centimeters. Find the area of the cover to the nearest hundredth centimeter.