## 16 - Light placed above the center of circular area

Problem 16
A light is to be placed above the center of a circular area of radius a. What height gives the best illumination on a circular walk surrounding the area? (When light from a point source strikes a surface obliquely, the intensity of illumination is

$I = \dfrac{k\sin \theta}{d^2}$

where $\theta$ is the angle of incidence and $d$ the distance from the source.)

## 35-36 Time Rates: Lengthening of shadow and movement of its tip in 3D space

Problem 35
An arc light hangs at the height of 30 ft above the center of a street 60 ft wide. A man 6 ft tall walks along the sidewalk at the rate of 4 ft/sec. How fast is his shadow lengthening when he is 40 ft up the street?

Problem 36
In Problem 35, how fast is the tip of the shadow moving?

## 17-18 Rate of shadow in the wall of a building

Problem 17
A light is placed on the ground 30 ft from a building. A man 6 ft tall walks from the light toward the building at the rate of 5 ft/sec. Find the rate at which the length of his shadow is changing when he is 15 ft from the building.

Problem 18
Solve Problem 17, if the light is 10 ft above the ground.

## 15-16 Movement of shadow from light at eye level

Problem 15
A light at eye level stands 20 ft from a house and 15 ft from the path leading from the house to the street. A man walks along the path at 6 ft per sec. How fast does his shadow move along the wall when he is 5 ft from the house?

Problem 16
In Problem 15, when the man is 5 ft from the house, find the time-rate of change of that portion of his shadow which lies on the ground.

## 43 - 45 Solved problems in maxima and minima

Problem 43
A ship lies 6 miles from shore, and opposite a point 10 miles farther along the shore another ship lies 18 miles offshore. A boat from the first ship is to land a passenger and then proceed to the other ship. What is the least distance the boat can travel?

Problem 44
Two posts, one 8 feet high and the other 12 feet high, stand 15 ft apart. They are to be supported by wires attached to a single stake at ground level. The wires running to the tops of the posts. Where should the stake be placed, to use the least amount of wire?

Problem 45
A ray of light travels, as in Fig. 39, from A to B via the point P on the mirror CD. Prove that the length (AP + PB) will be a minimum if and only if α = β.

## 06 Review Problem - Area illuminated by light placed in front of the globe

Problem 6
A light is placed 5 ft. from the center of the globe 3 ft. in diameter. Find the area of the illuminated portion.