Mrs Dundee, an Australian farmer, has four silos on her farm in which she stores fertiliser, as shown in the photograph and diagram below. The silos are cylindrical with a roof section. Fertiliser is only stored in the cylindrical part of the silos. The following formula and conversion rates may be used: Volume of a cylinder = π × (radius)² × height; using π = 3.142 1 m³ = 1 000 ℓ 1 kg = 1 litre 1 gallon = 3.7 litres 1. Calculate the diameter of a silo if the volume of the cylindrical part is 60 m³. 2. Calculate the total maximum capacity (in gallons) of the four silos. 3. After fertilising all her main fields, Mrs Dundee wants to use the remaining fertiliser for a wheat field, which is 1 055 acres in size. The capacity readings of the four silos are as follows: - Silo 1: 15% full - Silo 2: ½ full - Silos 3 and 4: empty Verify, showing ALL calculations, whether she will have enough fertiliser left in her silos for the wheat field if the spread rate is 22.65 kg of fertiliser per acre.

NSC Mathematical Literacy Representing, interpreting and analysing data: Mrs Dundee, an Australian farmer

Mrs Dundee, an Australian farmer, has four silos on her farm in which she stores fertiliser, as shown in the photograph and diagram below - NSC Mathematical Literacy - Question 5 - 2016 - Paper 2

Question 5

Mrs-Dundee,-an-Australian-farmer,-has-four-silos-on-her-farm-in-which-she-stores-fertiliser,-as-shown-in-the-photograph-and-diagram-below-NSC Mathematical Literacy-Question 5-2016-Paper 2.png

Mrs Dundee, an Australian farmer, has four silos on her farm in which she stores fertiliser, as shown in the photograph and diagram below. The silos are cylindrical ... show full transcript

Worked Solution & Example Answer:Mrs Dundee, an Australian farmer, has four silos on her farm in which she stores fertiliser, as shown in the photograph and diagram below - NSC Mathematical Literacy - Question 5 - 2016 - Paper 2

Step 1

Calculate the diameter of a silo if the volume of the cylindrical part is 60 m³.

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Answer

Using the volume formula for a cylinder, we have:

ext{Volume} = ext{π} imes ( ext{radius})^2 imes ext{height}

Here, the volume is 60 m³ and the height is 7.35 m. Therefore:

60 = 3.142 imes ( ext{radius})^2 imes 7.35

Rearranging for radius:

( ext{radius})^2 = rac{60}{3.142 imes 7.35}

Calculating:

ext{radius}^2 = 2.598111173

Thus, the radius is:

ext{radius} = ext{√}(2.598111173) \\ = 1.611865743 ext{ m}

The diameter, which is twice the radius, is:

ext{diameter} = 2 imes 1.611865743 = 3.223731486 ext{ m}

Therefore, the diameter of the silo is approximately 3.22 m.

Step 2

Calculate the total maximum capacity (in gallons) of the four silos.

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Answer

Since each silo has a volume of 60 m³, the total volume for four silos is:

ext{Total volume} = 4 imes 60 = 240 ext{ m}^3

To convert this to gallons:

1 ext{ m}^3 = 1,000 ext{ ℓ} ightarrow 240 ext{ m}^3 = 240,000 ext{ ℓ}

Now, converting litres to gallons:

240,000 ext{ ℓ} imes rac{1 ext{ gallon}}{3.7 ext{ ℓ}} \ = 64,864.86 ext{ gallons}

Thus, the total maximum capacity of the four silos is approximately 64,865 gallons.

Step 3

Verify, showing ALL calculations, whether she will have enough fertiliser left in her silos for the wheat field if the spread rate is 22.65 kg of fertiliser per acre.

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Answer

The total amount of fertiliser left in the silos is calculated as follows:

Silo 1: 15% of 60 m³ = 9 m³ (1 m³ = 1,000 kg) = 9,000 kg
Silo 2: 50% of 60 m³ = 30 m³ = 30,000 kg
Silos 3 and 4: 0 kg

Total fertiliser in silos:

ext{Total fertiliser} = 9,000 + 30,000 = 39,000 ext{ kg}

For a field of 1,055 acres:

ext{Fertiliser needed} = 1,055 imes 22.65 ext{ kg/acre} = 23,895.75 ext{ kg}

Since 39,000 kg > 23,895.75 kg, she will have enough fertiliser for the wheat field.

Mrs Dundee, an Australian farmer, has four silos on her farm in which she stores fertiliser, as shown in the photograph and diagram below - NSC Mathematical Literacy - Question 5 - 2016 - Paper 2

Worked Solution & Example Answer:Mrs Dundee, an Australian farmer, has four silos on her farm in which she stores fertiliser, as shown in the photograph and diagram below - NSC Mathematical Literacy - Question 5 - 2016 - Paper 2

Calculate the diameter of a silo if the volume of the cylindrical part is 60 m³.

Calculate the total maximum capacity (in gallons) of the four silos.

Verify, showing ALL calculations, whether she will have enough fertiliser left in her silos for the wheat field if the spread rate is 22.65 kg of fertiliser per acre.

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