A solid cylinder has a radius of 10 mm and a height of 45 mm. (a) Draw a sketch of the net of the surface of the cylinder and write its dimensions on the sketch. (b) Calculate the volume of the cylinder. Give your answer in terms of π. (c) A sphere has the same volume as the cylinder. Find the surface area of the sphere. Give your answer in terms of π.

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A solid cylinder has a radius of 10 mm and a height of 45 mm - Leaving Cert Mathematics - Question Question 1 - 2013

Question Question 1

A solid cylinder has a radius of 10 mm and a height of 45 mm. (a) Draw a sketch of the net of the surface of the cylinder and write its dimensions on the sketch. (... show full transcript

Worked Solution & Example Answer:A solid cylinder has a radius of 10 mm and a height of 45 mm - Leaving Cert Mathematics - Question Question 1 - 2013

Step 1

Draw a sketch of the net of the surface of the cylinder and write its dimensions on the sketch.

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Answer

To draw the net of a cylinder, visualize it as consisting of two circles (the top and bottom bases) and a rectangle (the lateral surface). The circles will have a radius of 10 mm, which indicates their diameter is 20 mm.

The rectangle will have a height of 45 mm and a width equal to the circumference of the base, calculated as:

$2 \pi r = 2 \pi (10) = 20\pi \approx 62.8 \text{ mm}$

Thus, the net includes:

Two circles (each with a diameter of 20 mm, and a radius of 10 mm)
One rectangle (with height 45 mm and width 62.8 mm)

Step 2

Calculate the volume of the cylinder. Give your answer in terms of π.

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Answer

The volume of a cylinder is calculated using the formula:

$V = \pi r^2 h$

Substituting the known values:

Radius, $r = 10 \text{ mm}$
Height, $h = 45 \text{ mm}$

We get:

$V = \pi (10)^2 (45) = \pi (100)(45) = 4500\pi \text{ mm}^3$

Step 3

Find the surface area of the sphere. Give your answer in terms of π.

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Answer

Since the sphere has the same volume as the cylinder, we use the volume of the sphere formula:

$\frac{4}{3} \pi r^3 = 4500\pi$

Equating the volumes:

$\frac{4}{3} r^3 = 4500\implies r^3 = 4500 \times \frac{3}{4} = 3375$

Solving for $r$ :

$r = \sqrt[3]{3375} = 15 \text{ mm}$

Now, using this radius, we can find the surface area (A) of the sphere:

$A = 4\pi r^2 = 4\pi (15)^2 = 4\pi (225) = 900\pi \text{ mm}^2$

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A solid cylinder has a radius of 10 mm and a height of 45 mm - Leaving Cert Mathematics - Question Question 1 - 2013

Worked Solution & Example Answer:A solid cylinder has a radius of 10 mm and a height of 45 mm - Leaving Cert Mathematics - Question Question 1 - 2013

Draw a sketch of the net of the surface of the cylinder and write its dimensions on the sketch.

Calculate the volume of the cylinder. Give your answer in terms of π.

Find the surface area of the sphere. Give your answer in terms of π.

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