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 1.3 kg = 1 litre 1 gallon = 3.7 litres 5.1.1 Calculate the diameter of a silo if the volume of the cylindrical part is 60 m³. 5.1.2 Calculate the total maximum capacity (in gallons) of the four silos. 5.1.3 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: - Silo 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.

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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

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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

To find the diameter of a silo, we start with the formula for the volume of a cylinder:

$V = ext{π} imes r^2 imes h$

Setting the volume, V, equal to 60 m³ and substituting in the height (7.35 m), we get:

$60 = 3.142 imes r^2 imes 7.35$

Rearranging the equation:

$r^2 = \frac{60}{3.142 imes 7.35}$

Calculating the values:

$r^2 = \frac{60}{23.0727} \approx 2.598111173$

Taking the square root gives us:

$r \approx 1.611865743$

Thus, the diameter, which is twice the radius, is:

$d \approx 2 \times 1.611865743 \approx 3.223730486$

Therefore, the diameter is approximately 3.22 m.

Step 2

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

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Answer

The total capacity of one silo in m³ is 60 m³. Thus, for four silos:

$Total\ Volume = 4 \times 60 = 240\ m³$

Now, converting this to gallons using the conversion rate:

$1\ m³ = 1 000\ ℓ \\ 1\ gallon = 3.7\ ℓ$

First, convert m³ to litres:

$240\ m³ = 240 \times 1,000 = 240,000\ ℓ$

Next, convert litres to gallons:

$Total\ gallons = \frac{240,000}{3.7} \approx 64,864.86\ gallons$

Therefore, the total maximum capacity of the four silos is approximately 64,864.86 gallons.

Step 3

After fertilising all her main fields, verify if she has enough fertiliser

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Answer

To find out if Mrs. Dundee has enough fertiliser, we first need to calculate the amount needed for a wheat field of 1,055 acres, using the spread rate of 22.65 kg/acre:

$Fertiliser\ needed = 1,055\ acres \times 22.65\ kg/acre \approx 23,895.75\ kg$

Now converting kg to litres:

$1.3\ kg = 1\ litre \\ \therefore\ 23,895.75\ kg = 23,895.75\ ℓ$

The total volume of fertiliser in the silos is given by their capacity readings:

Silo 1: 15% of 60 m³ = 9 m³
Silo 2: 30 m³
Silos 3 and 4: 0 m³

Total volume in m³:

$Total = 9 + 30 + 0 + 0 = 39\ m³$

Converting to litres:

$39\ m³ = 39\times 1,000 = 39,000\ ℓ$

Since 39,000 ℓ > 23,895.75 ℓ, Mrs. Dundee 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

After fertilising all her main fields, verify if she has enough fertiliser

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