The intensity, I, of light, in lumens, passing through the glass of a pair of sunglasses if given by the equation I(x) = I^o(0.8)^x, where x is the thickness of the glass, in millimeters, and I^o is the intensity of light entering the glasses. How thick should the glass be so that it will block 25% of the light entering the sunglasses?
Accurate answer?
To find the thickness of the glass that will block 25% of the light entering the sunglasses, we need to set up the equation and solve for x.
Given:
I(x) = I^o(0.8)^x (1)
We want to find the value of x where I(x) = 0.25 * I^o.
Substitute 0.25 * I^o for I(x) in equation (1):
0.25 * I^o = I^o(0.8)^x
Divide both sides of the equation by I^o to cancel out the I^o terms:
0.25 = 0.8^x
To solve for x, take the logarithm of both sides of the equation:
log(0.25) = log(0.8^x)
Use the logarithmic properties to bring down the exponent:
log(0.25) = x * log(0.8)
Solve for x by dividing both sides of the equation by log(0.8):
x = log(0.25) / log(0.8)
Using a calculator, evaluate log(0.25) and log(0.8) and divide them to find x.
To find the thickness of the glass that will block 25% of the light entering the sunglasses, we need to solve the equation I(x) = I^o(0.8)^x for x, where I(x) represents the intensity of light passing through the glass at thickness x.
Given that we want to block 25% of the light, the remaining intensity transmitting through the glass will be 75% of I^o. We can express this mathematically as 0.75I^o.
Setting I(x) = 0.75I^o in the equation I(x) = I^o(0.8)^x, we can solve for x.
0.75I^o = I^o(0.8)^x
Dividing both sides by I^o, we get:
0.75 = 0.8^x
To solve for x, we need to take the logarithm of both sides of the equation. You can use either the natural logarithm (ln) or the logarithm base 10 (log) to solve this equation. Let's use the natural logarithm.
ln(0.75) = ln(0.8^x)
Using the logarithmic property, ln(a^b) = b * ln(a), we can rewrite the equation as:
ln(0.75) = x * ln(0.8)
To solve for x, divide both sides of the equation by ln(0.8):
x = ln(0.75) / ln(0.8)
Using a calculator, we can find the approximate value of x:
x ≈ -0.226 / -0.223 = 1.015
Therefore, the glass should be approximately 1.015 millimeters thick to block 25% of the light entering the sunglasses.
We want
I^o(.25) = I^o(.8)^x
.25 = .8^x
ln.25 = x*ln.8
x = ln.25/ln.8 = 6.21