The intensity I of an x-ray after it has passed through a material that is x centimeters thick is given by I = I0e^–kx, where I0 is the initial intensity of the x-ray and k is a number that depends on the material. The constant k for copper is 3.2. Find the thickness of copper necessary so that the intensity of an x-ray after passing through the copper is 25% of the original intensity. Round to the nearest tenth.

Question

The intensity I of an x-ray after it has passed through a material that is x centimeters thick is given by I = I0e^–kx, where I0 is the initial intensity of the x-ray and k is a number that depends on the material. The constant k for copper is 3.2. Find the thickness of copper necessary so that the intensity of an x-ray after passing through the copper is 25% of the original intensity. Round to the nearest tenth.

Answer 1

.25 = e^-3.2x

so
ln .25 = -3.2 x

x = -ln .25/3.2

Answer 2

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To find the thickness of copper necessary for an x-ray to have 25% of its original intensity, we need to solve the equation:

0.25 = e^(-3.2x)

Taking the natural logarithm (ln) of both sides, we get:

ln(0.25) = ln(e^(-3.2x))

Applying the power rule of logarithms, we simplify further:

ln(0.25) = -3.2x

Now we can solve for x by dividing both sides by -3.2:

x = ln(0.25) / -3.2

Calculating this, we find:

x ≈ -0.2877 / -3.2

Rounding to the nearest tenth, the thickness of copper necessary is approximately 0.1 centimeters.

Answer 3

To find the thickness of copper necessary for the x-ray intensity to be 25% of the original intensity, we can use the equation I = I0e^(-kx).

Given:
Initial intensity, I0 = 1 (since it is a relative value)
Intensity after passing through copper, I = 0.25 (25% of I0)
Constant for copper, k = 3.2

Plugging in the given values into the equation, we have:
0.25 = 1 * e^(-3.2x)

To solve for x, we can take the natural logarithm (ln) of both sides:
ln(0.25) = ln(e^(-3.2x))

Using the property of logarithms, ln(e^a) = a, we simplify the equation to:
ln(0.25) = -3.2x

To isolate x, divide both sides of the equation by -3.2:
x = ln(0.25) / -3.2

Using a calculator, the value of ln(0.25) is approximately -1.3863.

Substituting this value into the equation, we have:
x = (-1.3863) / -3.2

Calculating this, we find:
x ≈ 0.4326

Rounded to the nearest tenth, the thickness of copper necessary is approximately 0.4 centimeters.

Answer 4

To find the thickness of copper necessary so that the intensity of an x-ray after passing through the copper is 25% of the original intensity, we need to solve the equation I = I0e^(-kx) for x.

Given:
I = 0.25 * I0 (25% of the original intensity)
k = 3.2 (constant for copper)

Substituting these values into the equation, we get:
0.25 * I0 = I0 * e^(-3.2x)

Next, divide both sides of the equation by I0:
0.25 = e^(-3.2x)

Now, take the natural logarithm (ln) of both sides of the equation to solve for x:
ln(0.25) = ln(e^(-3.2x))

Using the property of logarithms, ln(e^(-3.2x)) simplifies to -3.2x:
ln(0.25) = -3.2x

Now, divide both sides of the equation by -3.2:
x = ln(0.25) / -3.2

Using a calculator, we find that ln(0.25) is approximately -1.386:

x = -1.386 / -3.2
x ≈ 0.43375

Rounding to the nearest tenth, the thickness of copper necessary is approximately 0.4 centimeters.