To test an individual's use of a certain mineral, a researcher injects a small amount of a radioactive form of that mineral into the person's bloodstream. The mineral remaining in the bloodstream is measured each day for several days. Suppose the amount of the mineral remaining in the bloodstream (in miligrams per cubic centimeter) t days after the initial injection is approximated by \( \mathrm{C}(\mathrm{t})=\frac{1}{2}(4 \mathrm{t}+1)^{-1 / 2} \). Find and interpret the rate of change of the mineral level with respect to time for (a) 0 days, (b) 3 days, and (c) 5.5 days. (d) Are the slopes of the tangent lines to C always positive or always negative? What does this indicate about the rate of change of the minerall level? (a) The rate of change of the mineral level with respect to time for 0 days is -1 milligrams per cubic centimeter per day (Round to two decimal places as needed.)

To evaluate the rate of change of the mineral level with respect to time, we first need to find the derivative of the function \( \mathrm{C}(t) \). Using the formula provided: \[ \mathrm{C}(t) = \frac{1}{2}(4t + 1)^{-1/2} \] We differentiate this using the chain rule: \[ C'(t) = \frac{d}{dt} \left(\frac{1}{2}(4t + 1)^{-1/2}\right) = \frac{1}{2} \cdot -\frac{1}{2}(4t + 1)^{-3/2} \cdot 4 = -\frac{2}{(4t + 1)^{3/2}} \] Now, we can compute the value of the derivative \( C'(t) \) at the specified times: (a) For \( t = 0 \): \[ C'(0) = -\frac{2}{(4(0) + 1)^{3/2}} = -\frac{2}{1^{3/2}} = -2 \text{ mg/cm}^3/\text{day} \] (b) For \( t = 3 \): \[ C'(3) = -\frac{2}{(4(3) + 1)^{3/2}} = -\frac{2}{13^{3/2}} \approx -0.554 \text{ mg/cm}^3/\text{day} \] (c) For \( t = 5.5 \): \[ C'(5.5) = -\frac{2}{(4(5.5) + 1)^{3/2}} = -\frac{2}{23^{3/2}} \approx -0.424 \text{ mg/cm}^3/\text{day} \] (d) The slopes of the tangent lines to \( C(t) \) are always negative, as evidenced by our calculations; this indicates that the rate of change of the mineral level is continually decreasing over time. That means as time progresses, the amount of the mineral in the bloodstream diminishes. Now that we have the results, let’s summarize them: - (a) \( -2 \) mg/cm³/day - (b) \( \approx -0.55 \) mg/cm³/day - (c) \( \approx -0.42 \) mg/cm³/day The general trend of the mineral level in the bloodstream is declining, which is typical for substances that are eventually eliminated from the body.