High School

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------------------------------------------------ Amanda mixes fruit concentrate with water in the ratio 1:7 to make 8 litres of drinkable juice. She uses a cylindrical drum with a diameter of 60 cm and fills it to a height of 89 cm. She then pours the juice into 0.5-litre bottles and sells it to a school tuck shop.

3.1 Determine the volume of drinkable juice that can be contained in the drum.
Volume of a cylinder = π × radius² × height

3.2 How many litres of fruit concentrate does she need in order to make a drum of drinkable juice?

3.3 How many bottles can she fill up with drinkable juice from 1 drum?

3.4 Amanda plans to expand her sales to other school tuck shops. She analyses the total sales from 6 other tuck shops and works out the possibility of taking over the markets. Her observations are as follows:

| Tuck shop | 1 | 2 | 3 | 4 | 5 | 6 |
|-----------------|----|----|----|----|----|----|
| Amanda's juice | 13 | 17 | 33 | 12 | 11 | 21 |
| Other juices | 22 | 14 | 20 | 11 | 11 | 17 |

What is the probability that she will take over the market for the 6 schools?

Answer :

Let's address each part of the question step-by-step.

3.1 Determine the volume of drinkable juice that can be contained in the drum.

The formula for the volume of a cylinder is given by:

[tex]\text{Volume} = \pi \times \text{radius}^2 \times \text{height}[/tex]

Given:

  • Diameter of the drum = 60 cm. Therefore, the radius = 30 cm (since radius is half of the diameter).
  • Height of the drum = 89 cm.

Plug these values into the formula:

[tex]\text{Volume} = \pi \times (30)^2 \times 89[/tex]

[tex]\text{Volume} = \pi \times 900 \times 89[/tex]

[tex]\text{Volume} \approx 3.14 \times 900 \times 89[/tex]

[tex]\text{Volume} \approx 251,370 \text{ cubic centimeters}[/tex]

To convert cubic centimeters to liters, use the conversion factor: 1 liter = 1,000 cubic centimeters.

[tex]\text{Volume in liters} = \frac{251,370}{1,000}[/tex]

[tex]\text{Volume in liters} \approx 251.37 \text{ liters}[/tex]

So, the drum can contain approximately 251.37 liters of drinkable juice.

3.2 How many liters of fruit concentrate does she need in order to make a drum of drinkable juice?

Amanda mixes fruit concentrate with water in the ratio 1:7. So, for every 8 liters of drinkable juice.

Since the total volume of drinkable juice in the drum is approximately 251.37 liters, the concentrate needed is:

[tex]\frac{1}{8} \times 251.37[/tex]

[tex]\approx 31.42 \text{ liters of concentrate}[/tex]

3.3 How many bottles can she fill up with drinkable juice from 1 drum?

Each bottle holds 0.5 liters of juice. Therefore, the number of full 0.5-liter bottles Amanda can fill is:

[tex]\frac{251.37}{0.5}[/tex]

[tex]\approx 502.74 \text{ bottles}[/tex]

Amanda will be able to fill approximately 502 full bottles, with a little juice remaining.

3.4 Probability that she will take over the market for the 6 schools.

Amanda's total sales are found by adding the sales figures from all 6 schools:

13 + 17 + 33 + 12 + 11 + 21 = 107

The sales of other juices are:

22 + 14 + 20 + 11 + 11 + 17 = 95

To determine the probability of Amanda taking over the market, consider a scenario where Amanda's sales are greater than those of other juices.

  • Amanda takes over the market if her sales are more than half of the total sales (Amanda's sales + Other Juices Sales):

Total sales:

[tex]107 + 95 = 202[/tex]

Probability Amanda's sales are dominant:

[tex]\text{Probability} = \frac{107}{202} \approx 0.5297[/tex]

The probability that Amanda will take over the market is roughly 52.97% when rounded to two decimal places.

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