The area, A mm², of a bacterial culture growing in milk, t hours after midday, is given by A = 20e^{1.5t}, t > 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 Trigonometric Equations: 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 > 0 (a) Write down the area of the culture at midday - Edexcel - A-Level Maths Pure - Question 5 - 2012 - Paper 6

Question 5

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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 > 0 (a) Write down the area of the culture at midday.... 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 > 0 (a) Write down the area of the culture at midday - Edexcel - A-Level Maths Pure - Question 5 - 2012 - Paper 6

Step 1

Write down the area of the culture at midday.

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Answer

To find the area of the culture at midday, we need to substitute t = 0 into the equation.

a = 20e^{1.5(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

The area at midday is 20 mm². Therefore, we want to find t such that:

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

Using the original equation:

$40 = 20e^{1.5t}$

Dividing both sides by 20:

$2 = e^{1.5t}$

Taking the natural logarithm of both sides gives:

$\ln(2) = 1.5t$

Solving for t:

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

Calculating t:

$\ln(2) \approx 0.693$

So,

$t = \frac{0.693}{1.5} \approx 0.462$

To convert this time into minutes:

$0.462 \times 60 \approx 27.72 \text{ minutes}$

Rounding to the nearest minute, we find that the time is 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 > 0 (a) Write down the area of the culture at midday - Edexcel - A-Level Maths Pure - Question 5 - 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 > 0 (a) Write down the area of the culture at midday - Edexcel - A-Level Maths Pure - Question 5 - 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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