A supermarket has been investigating how long customers have to wait at the checkout. During any half hour period, the percentage, P%, of customers who wait for less than r minutes, can be modelled by P = 100(1 - e^{-kr}), where k is a constant. (a) If 50% of customers wait for less than 3 minutes, determine the value of k. (b) Calculate the percentage of customers who wait for 5 minutes or longer.

Scottish Highers Maths Exponential and Logs: A supermarket has been investiga

A supermarket has been investigating how long customers have to wait at the checkout - Scottish Highers Maths - Question 11 - 2018

Question 11

A supermarket has been investigating how long customers have to wait at the checkout. During any half hour period, the percentage, P%, of customers who wait for les... show full transcript

Worked Solution & Example Answer:A supermarket has been investigating how long customers have to wait at the checkout - Scottish Highers Maths - Question 11 - 2018

Step 1

If 50% of customers wait for less than 3 minutes, determine the value of k.

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Answer

To find the value of k, we start with the formula:

$P = 100(1 - e^{-kr})$

Substituting the values given:

$50 = 100(1 - e^{-3k})$

Dividing both sides by 100:

$0.5 = 1 - e^{-3k}$

Rearranging gives us:

$e^{-3k} = 1 - 0.5 = 0.5$

Next, we take the natural logarithm of both sides:

$-3k = ext{ln}(0.5)$

This simplifies to:

$k = -\frac{1}{3} \text{ln}(0.5)$

Calculating this value, we find:

$k \approx 0.231$

Step 2

Calculate the percentage of customers who wait for 5 minutes or longer.

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Answer

We need to determine the percentage of customers who wait for 5 minutes or longer, which can be calculated using the formula:

$P = 100(1 - e^{-5k})$

Substituting k with approximately 0.231:

$P = 100(1 - e^{-5 \cdot 0.231})$

Calculating the exponent:

$-5 \cdot 0.231 \approx -1.155$

Thus,

$P \approx 100(1 - e^{-1.155})$

Now, calculating e to the power of -1.155:

$e^{-1.155} \approx 0.316$

So,

$P \approx 100(1 - 0.316) \approx 68.4 \%$

This means roughly 68.4% of customers wait for 5 minutes or longer.

A supermarket has been investigating how long customers have to wait at the checkout - Scottish Highers Maths - Question 11 - 2018

Worked Solution & Example Answer:A supermarket has been investigating how long customers have to wait at the checkout - Scottish Highers Maths - Question 11 - 2018

If 50% of customers wait for less than 3 minutes, determine the value of k.

Calculate the percentage of customers who wait for 5 minutes or longer.

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