Question

Strength (\(\sigma\)) of an implanted suture is given by the equation \[ \sigma = \sigma_0 - 2 \log_e(t) \quad {for} \quad t \geq 1, \] where \(\sigma_0\) represents the original strength and time \(t\) is measured in weeks. If the strength of the suture is 2 MPa at 4 weeks, what will be its strength (in MPa) at 8 weeks? (Rounded off to two decimal places)

Hint:

In problems involving logarithmic decay or growth, ensure that you correctly apply the natural logarithm (\(\log_e\)) and handle the constants appropriately.

Answer 1

Given the equation for the strength of the suture: \[ \sigma = \sigma_0 - 2 \log_e(t) \] We are provided with the information that the strength of the suture is 2 MPa at \(t = 4\) weeks: \[ \sigma = 2 \quad {when} \quad t = 4 \] Substituting these values into the equation: \[ 2 = \sigma_0 - 2 \log_e(4) \] First, calculate \(\log_e(4)\): \[ \log_e(4) \approx 1.386 \] Substitute this value into the equation: \[ 2 = \sigma_0 - 2 \times 1.386 \] \[ 2 = \sigma_0 - 2.772 \] Solving for \(\sigma_0\): \[ \sigma_0 = 2 + 2.772 = 4.772 \] Now, we want to find the strength of the suture at \(t = 8\) weeks: \[ \sigma = 4.772 - 2 \log_e(8) \] Next, calculate \(\log_e(8)\): \[ \log_e(8) \approx 2.079 \] Substitute this value into the equation: \[ \sigma = 4.772 - 2 \times 2.079 \] \[ \sigma = 4.772 - 4.158 \] \[ \sigma \approx 0.61 \, {MPa} \] Thus, the strength of the suture at 8 weeks is approximately \(0.61 \, {MPa}\).