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A patient is given a dose of medicine - Scottish Highers Maths - Question 13 - 2023

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A patient is given a dose of medicine. The concentration of the medicine in the patient's blood is modelled by $C_t = 11 e^{-0.005t}$ where: • $t$ is the time, in m... show full transcript

Worked Solution & Example Answer:A patient is given a dose of medicine - Scottish Highers Maths - Question 13 - 2023

Step 1

(a) Calculate the concentration of the medicine 30 minutes after the dose was given.

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Answer

To find the concentration of the medicine after 30 minutes, we can substitute t=30t = 30 into the formula:

Ct=11e−0.005×30C_t = 11 e^{-0.005 \times 30}

Calculating the exponent gives:

Ct=11e−0.15C_t = 11 e^{-0.15}

Using the value of e−0.15≈0.8607e^{-0.15} \approx 0.8607, we have:

Ct≈11×0.8607≈9.4687 mg/L,C_t \approx 11 \times 0.8607 \approx 9.4687 \text{ mg/L},

which can be rounded to approximately 9.4 mg/L.

Step 2

(b) Calculate the time taken for this dose of the medicine to become ineffective.

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Answer

To find the time taken for the concentration to fall to 0.66 mg/L, we set up the equation:

0.66=11e−0.005t0.66 = 11 e^{-0.005t}

Dividing both sides by 11:

e−0.005t=0.6611≈0.0600e^{-0.005t} = \frac{0.66}{11} \approx 0.0600

Taking the natural logarithm of both sides gives:

−0.005t=ln⁡(0.0600)-0.005t = \ln(0.0600)

And solving for tt:

t=−ln⁡(0.0600)0.005t = -\frac{\ln(0.0600)}{0.005}

Calculating this yields:

t≈−−2.8130.005≈562.6 minutes,t \approx -\frac{-2.813}{0.005} \approx 562.6\text{ minutes},

which rounds to approximately 563 minutes.

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