College Algebra Assignment 3

[ryan_kalle]

1. If f(a + b) = f(a) + f(b) for some function f, prove that f(0) = 0. (When proving, a and b CANNOT be assigned specific numerical values!)

2. If f(ab) = f(a) . f(b) for some function f, prove that f(1) = 1. (When proving, a and b CANNOT be assigned specific numerical values!)

3. The luminous intensity I, measured in candela (cd), of a 100 watt light bulb is 130 cd. The law of illumination states that

E = I / d2

where E is the illumination and d is the distance in meters to the light bulb. Suppose you hold a book 1 meter away from a 150 watt bulb and begin walking away from the bulb at a rate of 1 m/s.

a. Express E in terms of the time t in seconds, E(t), after you begin walking.
b. When will the illumination on the book be 1% of its original value?

My instructors latest assignment.

The first two have me scratching my head right now. Might have something to do with it being late night.

I think I got the third.
t = d/r
d= sqrt(I/E)
t = sqrt(I/E)/r

sqrt(130/1.3)/1 = 10

Brain is fried right now but this makes sense at the moment.

 

[mmm4444bot]

a. Express E in terms of the time t in seconds, E(t), after you begin walking.

I think I got [it]

t = d/r If d represents the distance from the bulb at time t, then this is not correct.

t = sqrt(I/E)/r This does not express E in terms of t. This expresses t in terms of E (I and r are constants).

Hi Ryan:

It looks like you're studying composite functions.

It's given that E(d) = I/d^2.

This expresses E in terms of d.

In order to express E in terms of t, instead, we need to find the function that expresses d in terms of t.

Look at the given information. We start 1 meter away from the bulb, and we move 1 meter away every second.

When t = 0, d = 1.

When t = 1, d = 2.

When t = 2, d = 3.

In other words, the distance from the bulb will always be one meter more than the number of elapsed seconds.

Therefore, the function (in this exercise) that expresses distance in terms of time is d(t) = t + 1.

For part 3(a), the requested composite function is E?d.

E[d(t)] = I/[d(t)^2]

Does this make sense? Can you simplify the composite above, to express E in terms of t?

E(t) = ?

For part 3(b), we want to know how many seconds it takes the illumination to go from E(0) to 0.01 times E(0).

From the simplified function above for E(t), it is clear that E(0) = I.

Therefore, 0.01 E(0) = 0.01 I.

By algebraic substitution and simplification, we arrive at the following equation, to solve for t.

0.01 = 1/(t + 1)^2

I welcome specific questions.

Cheers, Mark

 

[ryan_kalle]

Ok so I understand the 3rd part and I would like to thank Mark very much for his help.

Following is my work

1) f(a + b) = f(a) + f(b) Prove f(0) = 0
f(b) = f(a + b) - f(a)
f(0) = f(a + 0) - f(a)
f(0) = f(a) - f(a)
f(0) = 0

2) f(ab) = f(a) * f(b) prove f(1) = 1
f(b) = f(ab)/f(a)
f(1) = f(a*1)/f(a)
f(1) = f(a)/f(a)
f(1) = 1

3a) E = I/d^2 is an expression of E in terms of d, so E(d) = I/d^2.
To express E in terms of t we must calculate d. The initial distance is 1 at t = 0, at t = 1 the distance would be 2 and for t = 2 the distance would be 3, so d = t+1.

So then we just substitute.

E(t) = I/(t+1)^2

3b) The initial luminous intensity is 130cd representing 100% or 1.
.01 = 1/(t+1)^2
.01(t+1)^2 = 1
(t+1)^2 = 1/.01
t+1 = sqrt(100)
t = 9