MTH 151 H Calculus I Writing Assignment #2: Bulb Illumination

This document presents a detailed solution to a Calculus I writing assignment, creatively applying calculus principles to a real-world physics problem. The assignment focuses on determining the optimal height to hang a bulb above a round table to maximize illumination at the edges. The solution utilizes the "Inverse Square Law of Illuminance" and "Lambert's Cosine Law" from optics, formulating equations and employing differentiation to find the height that yields maximum illumination. The student applies the Pythagorean theorem and trigonometric functions to establish relationships between the bulb's height, the table's radius, and the oblique distance to the table's edge. By differentiating the illumination equation with respect to height and setting it to zero, the student derives the optimal height, demonstrating a strong understanding of calculus and its application to optimization problems. The final result, confirmed by the second derivative test, provides a clear and concise answer to the problem.

Object – I have a round table which I sometimes use for study purposes. I don’t like putting a
lamp on the table so want to fix a bulb just above the centre of the table. To fulfill the purpose, I
need to calculate the height to which I should hang the bulb so that the maximum illumination
reaches at the edges of the table.
(bulb)
l
θ h
(table)
r
Assumption – As part of physics study I know that some rules of optics are applicable in this
situation:
i. “Inverse Square Law of Illuminance” which states that if light rays incident on an object
obliquely, the illumination is proportional to the square of the distance.
ii. “Lambert's Cosine Law” which states that illumination is directly proportional to the
cosine of the angle of incidence on the surface.
Solution – Total illumination (I) of the lamp is going to be
I ∝ cos θ
I ∝ 1
l2
Total illumination will be
I ∝ cos θ
l2
I =k cos θ
l2 ----------------------------------------------------- equation 1

Here ‘k’ is a constant of proportionality.
Now, we observe the figure we see a right angle triangle
We also know that the Pythagoras theorem is applicable only to the right angle triangle.
According to which-
r2 +h2 =l2
Here –
r = radius of the table
h = height of the bulb
l = oblique distance between the light source and corner of the table
l
h
r
According to the above triangle
cos θ= h
l
cos θ= h
√r 2+ h2
Fromm equation 1, we have
I = k
(r2 +h2 )
h
√r2 +h2
I =k h
(r ¿ ¿2+h2)
3
2 ¿
----------------------------------- equation 2
We know that as we change the height the illumination changes. Now we differentiate equation 2
w.r.t height ‘h’
θ

dI
dh =
k h
(r ¿ ¿2+h2)
3
2 − 3
2
k ( h ) (2 h)
( r ¿¿ 2+ h2)
5
2 ¿
¿ --------------------------------- equation 3
For maxima or minima we put dI
dh = 0
k h
(r ¿ ¿2+ h2)
3
2 − 3
2
k ( h ) ( 2 h )
(r ¿¿ 2+ h2)
5
2 =0 ¿
¿
3 h2=r2+ h2
Finally we get –
h= r
√ 2
now to check the result for maxima or minima we put h= r
√2 in second derivative of equation 1
d2 x
d h2 =
k h
(r ¿ ¿2+ h2)
3
2 −3.2 k h
(r ¿¿ 2+h2)
5
2 − 5
2
h
(r ¿ ¿ 2+ h2)
7
2 ¿
¿
¿ ------------------------------
equation 4
Putting h= r
√ 2 in equation 4, we get
d2 x
d h2 < 0

Thus the height h= r
√2 to the maximum illumination.