Question

1. An 8-inch diameter (I.D.) pipe is filled to a depth equal to one-third of its diameter. What is the area in flow?

2. Find the area of a washer formed by two concentric circles whose chord outside the small circle is 10 cm.

3. A goat is tied to a corner of a 30 ft by 35 ft building. If the rope is 40 ft. long and the goat can reach 1 ft. farther than the rope length, what is the maximum area of the goat can cover?

I need solution pleaseeee

Answer 1

1. Let be the diameter and the radius. Here is a diagram

We need to calculate the blue area. To do so, we consider the area of a semicircle and subtract the areas in gray. First note that the height of the triangle in the middle is

.

With this we calculate the angle of the gray lines with respect to the horizontal,

.

Since the blue area is the area of the semicircle, minus the area of the triangle, minus the area of the two arcs, we have

We calculate as . Therefore,

.

Replacing we get .

2. Let be the inner and the outer radius. Then the area is

We have that the opposite side of the angle is . So we have the equations and . Now we have

Therefore,

.

3. We can consider that the length of the rope is . There are two situations

We evaluate both areas and get that the first one is and the second one . So the maximum is .