A flat machine part consists of two circular ends attached to a plate, as shown (diagram not to scale). The sides of the plate, HK and PQ, are tangential to each circle. The larger circle has centre A and radius 4 cm. The smaller circle has centre B and radius r cm. The length of |HK| is 8 cm and |AB| = 20/73 cm. (a) Find r, the radius of the smaller circle. (Hint: Draw BT || KH, T ∈ AH.) (b) Find the area of the quadrilateral ABKH. (c) (i) Find ∠HAP1, in degrees, correct to one decimal place. (ii) Find the area of the machine part, correct to the nearest cm².

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A flat machine part consists of two circular ends attached to a plate, as shown (diagram not to scale) - Leaving Cert Mathematics - Question 7 - 2015

Question 7

A flat machine part consists of two circular ends attached to a plate, as shown (diagram not to scale). The sides of the plate, HK and PQ, are tangential to each cir... show full transcript

Worked Solution & Example Answer:A flat machine part consists of two circular ends attached to a plate, as shown (diagram not to scale) - Leaving Cert Mathematics - Question 7 - 2015

Step 1

Find r, the radius of the smaller circle.

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Answer

To find the radius r of the smaller circle, we set up the equation involving the Pythagorean theorem. We have

$|AT|^2 + |BT|^2 = |AB|^2$

where:

|AT| = 3r (the distance from A to T, the tangential point),
|BT| = 8r (the distance from B to T), and
|AB| = (\frac{20}{73}) cm.

Substituting these values, we get:

$(3r)^2 + (8r)^2 = \left(\frac{20}{73} \right)^2$

This leads us to:

$9r^2 + 64r^2 = 29200/5329$ $73r^2 = 29200/5329$

We simplify to find:

$r^2 = 400$

Thus, the radius r of the smaller circle is:

$r = 20 \text{ cm}$ .

Step 2

Find the area of the quadrilateral ABKH.

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Answer

The area of the quadrilateral ABKH can be calculated using the formula:

$|ABKH| = |BKHT| + |ΔABT|$

We can calculate:

|BKHT| = height × base = 20 × 160
|ΔABT| = (\frac{1}{2} × |AB| × height = \frac{1}{2} × 20 × 60$$

Thus,

$|ABKH| = 20 × 160 + \frac{1}{2} × 20 × 60 = 8000 \text{ cm}^2$ .

Step 3

Find ∠HAP1, in degrees, correct to one decimal place.

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Answer

To find the angle ∠HAP1, we can use the definition of the tangent function:

$\tan(\angle HAP) = \frac{160}{60}$

This gives:

$\angle HAP = \tan^{-1}\left(\frac{160}{60}\right)$

Calculating this results in:

$\angle HAP \approx 69.4°$

Thus,

$\angle HAP1 = 138.9°$ .

Step 4

Find the area of the machine part, correct to the nearest cm².

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Answer

The total area of the machine part is the sum of the area of the larger sector and the smaller sector. Using the formula for the area of sectors, we calculate:

$\text{Area}(||HAP||) + 2\text{Area}(ABKH) + \text{Area}(KBQ)$

Calculating gives us:

$\text{Area} = \pi(80)\frac{221-1}{360} + 2 × 8000 + \pi(60)\frac{138.9}{360}$

This leads to:

$\text{Total Area} = 12845.55 + 16000 + 484.85$

Finally,

$\text{Total Area}= 28833.4 \text{ cm}^2 \approx 28833 \text{ cm}^2$ .

A flat machine part consists of two circular ends attached to a plate, as shown (diagram not to scale) - Leaving Cert Mathematics - Question 7 - 2015

Worked Solution & Example Answer:A flat machine part consists of two circular ends attached to a plate, as shown (diagram not to scale) - Leaving Cert Mathematics - Question 7 - 2015

Find r, the radius of the smaller circle.

Find the area of the quadrilateral ABKH.

Find ∠HAP1, in degrees, correct to one decimal place.

Find the area of the machine part, correct to the nearest cm².

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