A company has a spherical storage tank. The diameter of the tank is 12 m. (i) Write down the radius of the tank. Radius = (ii) Find the volume of the tank, correct to the nearest m³. (b) The company paints the outside curved surface of the spherical tank. (i) Find the curved surface area of the tank, correct to one decimal place. (ii) The curved surface is painted with a special paint. One litre of paint will cover 3.5 m². Find how many litres of paint are used, correct to the nearest litre. (iii) The paint is sold in 25 litre tins. Each tin costs €180. Find the total cost of the paint. (c) At another site the company has a differently-shaped tank with the same volume. This tank has hemispherical ends and a cylindrical mid-section of length h, as shown. The radius of each hemispherical end is 4.5 m. (i) Find the volume of one hemispherical end, correct to the nearest m³. (ii) Find the length, h, of the cylindrical section, correct to one decimal place.

Leaving Cert Mathematics Area & Volume: A company has a spherical storag

A company has a spherical storage tank - Leaving Cert Mathematics - Question 8 - 2015

Question 8

A company has a spherical storage tank. The diameter of the tank is 12 m. (i) Write down the radius of the tank. Radius = (ii) Find the volume of the tank, corre... show full transcript

Worked Solution & Example Answer:A company has a spherical storage tank - Leaving Cert Mathematics - Question 8 - 2015

Step 1

Write down the radius of the tank.

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Answer

The radius of the tank can be calculated by halving the diameter. Since the diameter is 12 m, the radius is:

ext{Radius} = rac{12}{2} = 6 ext{ m}

Step 2

Find the volume of the tank, correct to the nearest m³.

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Answer

The volume $V$ of a sphere is given by the formula:

V = \frac{4}{3} \pi r^3

Substituting the radius:

V = \frac{4}{3} \pi (6)^3 \\ = \frac{4}{3} \pi (216) \\ = 904.8 ext{ m}^3 \\ \approx 905 ext{ m}^3

Step 3

Find the curved surface area of the tank, correct to one decimal place.

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Answer

The curved surface area $A$ of a sphere is given by:

A = 4 \pi r^2

Using the radius:

A = 4 \pi (6)^2 \\ = 4 \pi (36) \\ = 452.4 ext{ m}^2 \\ \approx 452.4 ext{ m}^2 \\ \approx 452.4 ext{ m}^2

Step 4

Find how many litres of paint are used, correct to the nearest litre.

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Answer

To find the number of litres of paint needed, divide the curved surface area by the coverage of one litre:

\text{Litres} = \frac{452.4}{3.5} \\ \approx 129 ext{ litres (rounded to nearest litre)}

Step 5

Find the total cost of the paint.

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Answer

If each tin covers 25 litres, then the number of tins required is:

\text{Tins} = \frac{129}{25} \\ \approx 5.16 ext{ tins}

Since we need whole tins, round up to 6 tins. The total cost is:

\text{Total Cost} = 6 \times 180 = €1080

Step 6

Find the volume of one hemispherical end, correct to the nearest m³.

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Answer

The volume $V_h$ of a hemispherical end can be calculated as half the volume of a full sphere:

V_h = \frac{1}{2} \times \frac{4}{3} \pi r^3

dwhere ( r = 4.5 ) m:

V_h = \frac{1}{2} \times \frac{4}{3} \pi (4.5)^3\\ = \frac{1}{2} \times \frac{4}{3} \pi (91.125) \\ = 190 ext{ m}^3 \\ \approx 191 ext{ m}^3

Step 7

Find the length, h, of the cylindrical section, correct to one decimal place.

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Answer

The total volume $V$ of the tank must remain constant. Thus, the volume formula for the cylindrical section is:

V = 2V_h + \pi r^2 h

Substituting known values:

523 = 2(191) + \pi (4.5)^2 h \\ = 382 + 20.25 \pi h

Solving for ( h ):

523 - 382 = 20.25 \pi h \\ 141 = 20.25 \pi h \\ h = \frac{141}{20.25 \pi} \\ \approx 2 ext{ m}

A company has a spherical storage tank - Leaving Cert Mathematics - Question 8 - 2015

Worked Solution & Example Answer:A company has a spherical storage tank - Leaving Cert Mathematics - Question 8 - 2015

Write down the radius of the tank.

Find the volume of the tank, correct to the nearest m³.

Find the curved surface area of the tank, correct to one decimal place.

Find how many litres of paint are used, correct to the nearest litre.

Find the total cost of the paint.

Find the volume of one hemispherical end, correct to the nearest m³.

Find the length, h, of the cylindrical section, correct to one decimal place.

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