A solid cone has a radius of 5 cm and a vertical height of 12 cm, as shown. (a) Use the theorem of Pythagoras to work out the value of l, the slant height of the cone. (b) Work out the total surface area of the cone. Give your answer in cm², correct to one decimal place. (c) The net of this cone is made up of two parts, a circle and a sector. These are shown in the table below. Complete the table by filling in each of the four missing measurements. Give each value in cm, correct to one decimal place where necessary.

Junior Cycle Mathematics Trigonometry: A solid cone has a radius of 5 c

A solid cone has a radius of 5 cm and a vertical height of 12 cm, as shown - Junior Cycle Mathematics - Question 3 - 2018

Question 3

A solid cone has a radius of 5 cm and a vertical height of 12 cm, as shown. (a) Use the theorem of Pythagoras to work out the value of l, the slant height of the co... show full transcript

Worked Solution & Example Answer:A solid cone has a radius of 5 cm and a vertical height of 12 cm, as shown - Junior Cycle Mathematics - Question 3 - 2018

Step 1

Use the theorem of Pythagoras to work out the value of l

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Answer

To find the slant height, l, of the cone, we apply the Pythagorean theorem:

$l^2 = r^2 + h^2$

where:

$r = 5 \text{ cm}$ (radius)
$h = 12 \text{ cm}$ (height)

Substituting the values:

$l^2 = 5^2 + 12^2$ $l^2 = 25 + 144$ $l^2 = 169$

Taking the square root:

$l = \sqrt{169} = 13 \text{ cm}$

Step 2

Work out the total surface area of the cone

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Answer

The total surface area (TSA) of a cone is given by:

$TSA = \pi r l + \pi r^2$

Substituting the values we found:

$r = 5 \text{ cm}$
$l = 13 \text{ cm}$

Calculating each part:

Curved surface area: $\pi r l = \pi (5)(13) = 65\pi \text{ cm}^2$
Base area: $\pi r^2 = \pi (5)^2 = 25\pi \text{ cm}^2$

Now adding both areas:

$TSA = 65\pi + 25\pi = 90\pi \approx 282.7 \text{ cm}^2$

Rounded to one decimal place: TSA ≈ 282.7 cm².

Step 3

Radius of the circle

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Answer

Radius of circle = 5 cm

Step 4

Circumference =

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Answer

Circumference = $2\pi(5) \approx 31.4 \text{ cm}$

Step 5

Radius of sector =

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Answer

Radius of Sector = 13 cm

Step 6

Length of the arc =

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Answer

Length of Arc = 31.4 cm

A solid cone has a radius of 5 cm and a vertical height of 12 cm, as shown - Junior Cycle Mathematics - Question 3 - 2018

Worked Solution & Example Answer:A solid cone has a radius of 5 cm and a vertical height of 12 cm, as shown - Junior Cycle Mathematics - Question 3 - 2018

Use the theorem of Pythagoras to work out the value of l

Work out the total surface area of the cone

Radius of the circle

Circumference =

Radius of sector =

Length of the arc =

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