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Advanced Math / Nonlinear functions Difficulty: Medium

$f (t) = 500 {(0.5)}^{\frac{t}{12}}$

The function $f$ models the intensity of an X-ray beam, in number of particles in the X-ray beam, $t$ millimeters below the surface of a sample of iron. According to the model, what is the estimated number of particles in the X-ray beam when it is at the surface of the sample of iron?

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Explanation

Choice A is correct. It's given that the function $f$ models the intensity of an X-ray beam, in number of particles in the X-ray beam, $t$ millimeters below the surface of a sample of iron. When the X-ray beam is at the surface of the sample of iron, it is $0$ millimeters below the surface, so the value of $t$ is $0$ . Substituting $0$ for $t$ in the function $f (t) = 500 {(0.5)}^{\frac{t}{12}}$ yields $f (0) = 500 {(0.5)}^{\frac{0}{12}}$ . Since any positive number raised to the power of $0$ is equal to $1$ , it follows that $f left parenthesis 0 right parenthesis equals 500 left parenthesis 1 right parenthesis$ , or $f left parenthesis 0 right parenthesis equals 500$ . Therefore, the estimated number of particles in the X-ray beam at the surface of the sample of iron is $500$ .

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.