[-/1 Points] DETAILS MY NOTES LARCALC12 2.4.507.XP. ASK YOUR TEACHER PRACTICE ANOTHER The speed \( S \) of blood that is \( r \) centimeters from the center of an artery is given below, where \( C \) is a constant, \( R \) is the radius of the artery, and \( S \) is measured in centimeters per second. Suppose a drug is administered and tha artery begins to ollate at a rate of dR/dt. At a constant distance r, find the rate (in cm/s) at which s changes with respect to \( t \) for \( C=1.26 \times 10^{5}, R=1.3 \times 10^{-2} \), and \( d R / d t=2.0 \times 10^{-5} \). (Round your answer to 4 decimal places.) \[ \begin{array}{c} s=c\left(R^{2}-r^{2}\right) \\ \frac{d s}{d t}=\square \mathrm{cm} / \mathrm{s} \end{array} \]

We start with the given formula for the speed of the blood:   s = C (R² − r²) where:   C = 1.26×10⁵,   R = radius of the artery,   r = fixed distance from the center (constant),   and the rate of dilation is given by dR/dt. Since r is constant, differentiate s with respect to t:   ds/dt = C d/dt (R² − r²) = C (2R dR/dt − 0) = 2C R (dR/dt). Now substitute the given values:   C = 1.26×10⁵,   R = 1.3×10⁻² cm,   dR/dt = 2.0×10⁻⁵ cm/s. Plug these into the formula:   ds/dt = 2 × (1.26×10⁵) × (1.3×10⁻²) × (2.0×10⁻⁵). Step 1: Compute 2 × (1.26×10⁵):   2 × 1.26×10⁵ = 2.52×10⁵. Step 2: Multiply by R:   2.52×10⁵ × 1.3×10⁻² = 2.52 × 1.3 × 10^(5 − 2)     = 3.276 × 10³. Step 3: Multiply by dR/dt:   3.276×10³ × 2.0×10⁻⁵ = 3.276 × 2.0 × 10^(3 − 5)     = 6.552 × 10⁻². Thus, ds/dt ≈ 0.06552 cm/s, which when rounded to 4 decimal places is:   ds/dt ≈ 0.0655 cm/s. So the rate at which the speed s changes with respect to time t is 0.0655 cm/s.