The square ABCD has sides of length 7 cm. The vertices of the square PQRS lie on the perimeter of ABCD, as shown in the diagram, with |AQ| = 5 cm. Find the area of the square PQRS. The circles u and v represent two wheels that are free to rotate about their centres, as shown. The radius of u is 4 cm and the radius of v is 6 cm. (i) Find the length of the circumference of each circle. Give your answers in terms of π. (ii) The wheels u and v are in non-slip contact and therefore the rotation of one causes the other to rotate. Find the number of complete rotations wheel u makes if wheel v completes 100 rotations.

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The square ABCD has sides of length 7 cm - Leaving Cert Mathematics - Question 5 - 2018

Question 5

The square ABCD has sides of length 7 cm. The vertices of the square PQRS lie on the perimeter of ABCD, as shown in the diagram, with |AQ| = 5 cm. Find the area of t... show full transcript

Worked Solution & Example Answer:The square ABCD has sides of length 7 cm - Leaving Cert Mathematics - Question 5 - 2018

Step 1

Find the area of the square PQRS

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Answer

To find the area of square PQRS, we first calculate the area of square ABCD:

$ext{Area of ABCD} = 7 imes 7 = 49 ext{ cm}^2$

Next, we find the area of triangle AQP. The length |AQ| is given as 5 cm, and since PQRS is inside square ABCD, we can find the dimensions of AQ and PQ:

The height of triangle AQP can be calculated as the length of AD minus |AQ|, which is: $|AP| = 7 - 5 = 2 ext{ cm}$

The area of triangle AQP is:

$ext{Area of } AQP = \frac{1}{2} \times base \times height = \frac{1}{2} \times 5 \times 2 = 5 ext{ cm}^2$

Now, we can find the area of square PQRS:

$ext{Area of PQRS} = ext{Area of ABCD} - 4 \times ext{Area of AQP}$ $= 49 - 4 \times 5 = 29 ext{ cm}^2$

Step 2

Find the circumference of circles u and v

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Answer

For circle u:

The radius is 4 cm.
The circumference is given by:

$C_u = 2\pi r = 2\pi(4) = 8\pi ext{ cm}$

For circle v:

The radius is 6 cm.
The circumference is given by:

$C_v = 2\pi r = 2\pi(6) = 12\pi ext{ cm}$

Step 3

Find the number of complete rotations wheel u makes

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Answer

Since the wheels u and v are in non-slip contact, the distance traveled by each wheel is equal when one completes a certain number of rotations.

If wheel v completes 100 rotations:

The distance traveled by wheel v:

$D_v = 100 \times C_v = 100 \times 12\pi = 1200\pi ext{ cm}$

Now we find the number of rotations wheel u makes:

Let x be the number of complete rotations made by wheel u:

This distance is equal to:

$D_u = x \times C_u = x \times 8\pi$

Setting these equal gives:

$100 \times 12\pi = x \times 8\pi$

Divide each side by $\pi$ and solve for x:

$1200 = 8x$

Thus,

$x = \frac{1200}{8} = 150$

Therefore, wheel u makes 150 complete rotations.

The square ABCD has sides of length 7 cm - Leaving Cert Mathematics - Question 5 - 2018

Worked Solution & Example Answer:The square ABCD has sides of length 7 cm - Leaving Cert Mathematics - Question 5 - 2018

Find the area of the square PQRS

Find the circumference of circles u and v

Find the number of complete rotations wheel u makes

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