word problem help please

Question

the light intensity I (in lumens) at a depth of x feet in Lake Elizabeth in given by the formula log(I/12)=-0.025x.  a. Find the light intensity at a depth of 32 feet.    b.At what depth is the light intensity 6 lumens?

Answer 1

The light intensity , l, at x=32

 

 

log(l / 12) = -0.025(32)

log(l / 12) = -8 / 10

log(l / 12) = -4 / 5

 

This is a logarithmic equation.  The solution to a logarithm is the exponent of the base number 10.

 

(10)^-4/5 = l / 12

 

⁵ √(1 / 10⁴) = l / 12

 

 

Multiply both sides of equation by 12.

 

12 * ⁵ √(1 / 10⁴) = l

 

12 / ⁵ √(10⁴) = l

 

1.901 = l

 

 

The depth, x, at l=6

 

log(6 / 12) = -0.025x

log(1/2) = -0.025x

 

Divide both sides of equation by -0.025

 

 

log(12) / (-0.025) = x

 

12.04 = x

 

 

 

 

 

Answer 2

log(I/12) = -0.025x

 

a. Find the light intensity at a depth of 32 feet

 

10^log(I/12) = 10^-0.025x

 

I/12 = 10^-0.025x

I = 12*10^-0.025x

 

For x = 32 feet:

 

I = 12*10^-0.025*32

I = 12*10^-0.8

I = 1.9 lumens

 

b. At what depth is the light intensity 6 lumens?

 

Use the original formula given in the problem statement:

 

log(I/12) = -0.025x

log(6/12) = -0.025x

log(1/2) = -0.025x

log(1/2)/-0.025 = x

12 feet = x