The area, A mm², of a bacterial culture growing in milk, t hours after midday, is given by A = 20e^{1.5t}, \, t \geq 0 (a) Write down the area of the culture at midday. (b) Find the time at which the area of the culture is twice its area at midday. Give your answer to the nearest minute.

A-Level Edexcel Maths Pure Equation of a Straight Line: The area, A mm², of a bacterial

The area, A mm², of a bacterial culture growing in milk, t hours after midday, is given by A = 20e^{1.5t}, \, t \geq 0 (a) Write down the area of the culture at midday - Edexcel - A-Level Maths Pure - Question 4 - 2012 - Paper 6

Question 4

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The area, A mm², of a bacterial culture growing in milk, t hours after midday, is given by A = 20e^{1.5t}, \, t \geq 0 (a) Write down the area of the culture at mi... show full transcript

Worked Solution & Example Answer:The area, A mm², of a bacterial culture growing in milk, t hours after midday, is given by A = 20e^{1.5t}, \, t \geq 0 (a) Write down the area of the culture at midday - Edexcel - A-Level Maths Pure - Question 4 - 2012 - Paper 6

Step 1

Write down the area of the culture at midday.

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Answer

To determine the area of the culture at midday, we set the time, t, to 0.

Substituting into the equation:

$A = 20e^{1.5 imes 0} = 20e^0 = 20 \text{ mm}^2$

Thus, the area of the culture at midday is 20 mm².

Step 2

Find the time at which the area of the culture is twice its area at midday. Give your answer to the nearest minute.

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Answer

We need to find the time t when the area A is twice the area at midday:

$A = 2 \times 20 = 40 \text{ mm}^2$

Setting the equation for area equal to 40 mm²:

$40 = 20e^{1.5t}$

Dividing both sides by 20:

$2 = e^{1.5t}$

Taking the natural logarithm of both sides:

$\ln(2) = 1.5t$

Now solving for t:

$t = \frac{\ln(2)}{1.5}$

Using a calculator, we find:

$t \approx 0.4624 \text{ hours}$

To convert this to minutes, multiply by 60:

$0.4624 \times 60 \approx 27.744 \text{ minutes}$

Rounding to the nearest minute gives approximately 28 minutes.

The area, A mm², of a bacterial culture growing in milk, t hours after midday, is given by A = 20e^{1.5t}, \, t \geq 0 (a) Write down the area of the culture at midday - Edexcel - A-Level Maths Pure - Question 4 - 2012 - Paper 6

Worked Solution & Example Answer:The area, A mm², of a bacterial culture growing in milk, t hours after midday, is given by A = 20e^{1.5t}, \, t \geq 0 (a) Write down the area of the culture at midday - Edexcel - A-Level Maths Pure - Question 4 - 2012 - Paper 6

Write down the area of the culture at midday.

Find the time at which the area of the culture is twice its area at midday. Give your answer to the nearest minute.

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