Expert: Paul Klarreich - 12/2/2007

Question
Please help me with my calculus problems.
8. the amount of illumination is proportional to the intensity of the light source, inversely proportional to the square of the distance from the light source, and proportional to sin(a) where a is the angle at which the light strikes the surface. a rectangular room measures 10x24 ft with a 10ft cieling. determine the height at which the light should be placed to allow the corners of the floor to recieve maximum light.
so the equation is something like I=(ksina)/d^2.
and they also tell you that the light is 13 ft away from the corner. so i guess its in the middle of the room.

10. the line joining P and Q crosses the two parallel lines as shown in figure. the point R is d units from P. how far from Q should the point S be chosen so that the sum of the areas of the two shaded triangles is a minimum? So that the sum is a maximum?

figure:
___S_____Q_
   \    /
    \  /
     \/    
     /\
    /  \
   /    \
-------------
   P_____R
      d

Answer
Questioner:   Punjab
Category:  Calculus
Private:  No
 
Subject:  applying derivatives.
Question:  Please help me with my calculus problems.
8. the amount of illumination is proportional to the intensity of the light source, inversely proportional to the square of the distance from the light source, and proportional to sin(a) where a is the angle at which the light strikes the surface. a rectangular room measures 10x24 ft with a 10ft cieling. determine the height at which the light should be placed to allow the corners of the floor to recieve maximum light.
so the equation is something like I=(ksina)/d^2.
and they also tell you that the light is 13 ft away from the corner. so i guess its in the middle of the room.
>> A good observation.
..................................................
Hi, Punjab,

Here's what I have on your first one;  I'll see what I can do about the other and try to send a followup.

Let d = the distance from the bulb to a corner:  
Let the bulb be at height y.  

Then we have  d^2 = sqrt(y^2 + 13^2)

Also  sin a(alpha?) = y/d

     k y/d
I =  -----------
      d^2

         k y
I =  ------------------
     (y^2 + 13^2)^3/2

         ((y^2+13^2)^3/2)(1) - (y)(3/2(y^2+13^2)^1/2(2y))  
dI/dy = k -------------------------------------------------
                          (who cares)^3


         (y^2 + 13^2)^3/2 - (y)(3y(y^2 + 13^2)^1/2)  
dI/dy = k ---------------------------------------------
                          (who cares)^3


         (y^2 + 13^2)^1/2[y^2 + 13^2 - 3y^2]
dI/dy = k ------------------------------------
                          (who cares)^3


         (y^2 + 13^2)^1/2[13^2 - 2y^2]
dI/dy = k ------------------------------
                   (who cares)^3

Set 13^2 - 2y^2 = 0;

y^2 = 13^2/2

y = 13/sqrt(2)
=========================================================
I am not totally satisfied with the following, but it might be usable:

10. the line joining P and Q crosses the two parallel lines as shown in figure. the point R is d units from P. how far from Q should the point S be chosen so that the sum of the areas of the two shaded triangles is a minimum? So that the sum is a maximum?

figure:

  |<- x->|
___S__C___Q_
  \  |  /
   \ | / h
    \|/    
    /\A
   / | \ 1-h
  /  |  \
-------------
  P__D___R
     d
Assume that the distance between the // lines is 1.  We have two similar triangles, ASQ and ARP, in that correspondence.

Now the heights from A down and from A up are  h and 1-h, in that order.  And we have sides in proportion:
SQ   RP
--- = ---
AC   AD

x     d
--- = ---
h    1-h

Our areas are 1/2 base * height.  We can forget the 1/2, since we are max-min-ing.

A(top) = hx
A(bottom) = d(1 - h)

Total area = hx + d(1 - h)

Now use our proportion:

x     d
--- = -----
h    1 -h

x - xh = dh


x = dh + xh
       x
h = ------
   d + x
Subst:
A  = hx + d(1 - h)
     x^2          dx
A = ------ + d - -----
   d + x        d + x  

         x^2 - dx
A = d +  ----------
         d + x

Now differentiate:
       (d + x)(2x - d) - (x^2 - dx)(1)
dA/dx = -------------------------------
               (d + x)^2


       2dx + 2x^2 - d^2 - dx - x^2 + dx
dA/dx = --------------------------------
               (d + x)^2

       2dx + x^2 - d^2   
dA/dx = -----------------
           (d + x)^2


 2dx + x^2 - d^2   

x^2 + 2dx     = d^2

x^2 + 2dx + d^2 = 2d^2

(x + d)^2 = 2d^2

x + d = d sqrt(2)

x = - d + d sqrt(2)
 = d(-1 + 1.414) ~~ 0.414d

That's a min; I don't think there is a max.