Please show your working out and your answers clearly on the separate answer sheet. You have a map of the grounds but also read the instructions with each question.



1. At the coach and car park by Flagstaff Gate

At Blenheim Palace we count every coach as having 40 passengers and every car as having two.

Count the number of coaches and cars parked in the car park area by Flagstaff Gate and work out the percentage of visitors who come in each way:

a. by coach
b. by car


2. At the crossroads, with a stop watch, for 10 minutes

For this question please walk away from Flagstaff Gate, with the Gate behind you, towards the crossroads near to the train station.
Stand on the grass by the fingered sign-post on the left.

a. Draw up a frequency table to tally the number of people per vehicle passing through the cross-roads. (Think how you will deal with coaches!)

b. Time yourself for 10 minutes and tally your results into your table.

c. From your table find the modal, median and an estimate of the mean number of people per vehicle.

3. The narrow-gauge railway

The narrow-gauge railway links the Palace to the Pleasure Gardens. A firm favourite with children and adults alike, the railway operates daily throughout the season (from 14 February to 12 December) at half-hour intervals. The diesel locomotive is named after Sir Winston Churchill who was born at Blenheim Palace.

Join the queue for the train (it’s free to all visitors).

The length of the track is 800m.
Time how long the train takes to complete its journey to the Pleasure Gardens and thus work out an estimate of its average speed.
Now make the return journey on the train and work out the average speed for the return journey – is it the same as for the outward journey?

4. The Bridge

For this question turn left at the crossroads and follow the road heading towards Woodstock Gate. Half-way down this road stop to admire the Bridge across the lake, on your left.
When this bridge was built, in 1723, it was the largest unsupported single arch bridge in the country. Of course it was only designed to take carriages in those days – now it still copes comfortably with much heavier vehicles such as delivery lorries.

The central arch was the biggest in Britain when it was built. At water level what do you estimate the width of the arch to be?

i. 66 metres
ii. 44 metres
iii. 33 metres

5. At Woodstock Gate

Follow the road down, until you reach Woodstock Gate. When you reach the gate look up.

a. The date when the gate was built is given using numbers as the Romans used them: MDCCXXIII. Nowadays we use Arabic numbers: 1, 2, 3, etc. What is the date of the gate, in Arabic numbers?

b. The Palace takes its name from the Battle of Blenheim, which took place in 1704 and was a victory for the English and Dutch against the French in Germany. The victory proved a crucial one which changed the course of European history and Queen Anne ordered that a palace should be built for John Churchill, the leader of the allied troops, to show the nation's gratitude.

Can you write the date of the Battle of Blenheim in Roman numerals?

6. At the steep slope down from Woodstock Gate

Now take the turning on the left down a slope.

The horizontal distance travelled as you go down the slope is 150m, the vertical distance you descend is 15m.

a. What is the steepness of the gradient:
i. as a ratio in the form 1:n?
ii. as a percentage?

b. By using Pythagoras can you estimate the distance travelled along the road surface, as you walk down this slope?

7. At the small bridges over the water

As you reach the bottom of the slope you will cross a stream of water. Stop and look over the side of the first bridge.

a. Puzzle: 'Seven of what cross H two O here?'

Now walk in a few more metres to the second smaller bridge and look over the side.

b. Suggest a method you could use to estimate the speed of water passing beneath you.

c. Use your method to estimate the speed of the water in metres per second.

d. Convert your speed to kilometres per hour.

8. Stone wall

As you reach the bottom of the slope and cross over the bridges you will notice a wall on your right.

The dry-stone wall of Woodstock Park is said to have been the first park wall built in England. When the Marlboroughs took over in 1705, it was in a poor state of repair and was not rebuilt until after the 1st Duke's death; nor was it finished until 1729.

All things considered it has lasted well, but still calls for constant maintenance, which is becoming more difficult as the number of skilled masons capable of doing the repairs is sadly decreasing. When the wall was originally rebuilt, the work was carried out by masons William Townsend of Oxford and Bartholomew Peisley junior, the son of the master mason who built the Grand Bridge – their estimate for the work was £1,196 per mile.

The wall here is approximately 33 metres long. If it takes six men twelve days to build a wall 33 metres long, how long will it take two men to build a wall 165m long?

9. Ancient oak tree

As you carry on walking, passing a cottage on your left, you will reach a road junction with an ancient, 200-year-old oak tree at the junction – the tree has two benches at its base.

The 9th Duke of Marlborough planted no fewer than 465,000 trees between 1893 and 1919 but more recently, Blenheim Park has been badly hit by freak storms, which brought down many of the trees planted in the times of the 1st and 4th Dukes; and also by Dutch elm disease which in spite of every precaution, destroyed the two main avenues on the north and east of the Park.

a. By some of your team joining hands, reaching around the tree and then by stretching out the same distance in a line – estimate the circumference.

b. The usable timber (the saw log) is from ground level to the first branch.
Estimate this height and then, with the two measurements you have, work out the volume of usable timber. If oak timber is worth £600 per cubic metre, how valuable is this tree?

c. BUT the forester may decide not to cut it down – for which of the following reasons

i. It may fall on his head
ii. It is protected by law
iii. It would be a pity to ruin something which has lasted so long
iv. He has never ever cut down such a big tree
v. It would be impossible to cut down such a big tree

10. The Column of Victory

Having turned left at the tree junction you will follow a road by the side of the Lake. Look out for the rare breed of bird called Snow Geese – as these birds have naturally white plumage (with a black streak on their tails) it is unusual to see these birds so far south. Their white colour makes them vulnerable to attack by foxes. See how many you can spot as you look out over the Lake.

Just as the road levels off and you come level with the end of a pier, leave the road and walk onto the grassy bank to the Lake, on your left. Look out for marks in the ground which show that a medieval watermill was once on this spot. Can you see a flat area where there was once a water wheel? Can you also see the old water course which is now covered by grass?

Continue along this road until you come to a gate on your right, marked 'Please close gate' at a road junction. Look over the gate towards the Column of Victory.

Started five years after Marlborough's death, the Column of Victory was completed in 1730 at a cost of £3,000. This doric column is surmounted by eagles and includes a lead statue of the 1st Duke of Marlborough by the otherwise unknown craftsman, Robert Pit.

Many designs and proposals were put forward for the monument and at one time an obelisk, standing halfway along the Great Avenue was planned but a column was decided upon and its current position at the entrance of the Great Avenue was finally chosen by the 1st Duchess.

The Column of Victory

What do you estimate its height to be?
a. 45 metres
b. 35 metres
c. 60 metres

11. The Main Entrance to the Palace and the North Gate

Now turn through 180° and walk towards the front of the Palace and the beautiful wrought iron North Gate ahead. Stop when you get to the gates.

a. The view of the Palace from this point shows off its symmetry to great effect. But something on the building spoils the perfect symmetry – can you spot what it is?

b. On your answer sheet you have been given a plan of this part of the Palace. Can you find the following things from the given clues/descriptions and mark them on your plan?
i. Orbs of Victory. Mark with a letter O. (How many can you spot?)
ii. The Lion of England pulling the Cockerel of France to pieces in its forepaws. Mark with the letter L.
iii. Trophies of war? Mark with the letter T. (The hedge maze is based on this design).
iv. Two chained captives. Mark with the letter C.

c. What is unusual about the Roman numerals on the clock face on the left tower? Why do you think the face was deliberately made this way?

d. The extension to the Palace on the left of your view has a triangular face at roof level. Assume that this extension is cuboidal, measuring 30 metres wide, 25 metres deep and 10 metres high and that its walls are one metre thick. Can you work out the mass of stone used to build the walls (ignoring windows and doors), given that one cubic metre of stone has a mass of one tonne.

e. Now look at the gates themselves. Choose a section of the gate which illustrates symmetry and sketch this section (do not spend more than ten minutes on this task).

12. Area and Volume at the main Palace Gate (Flagstaff Gate)

When you leave the North Gate turn left and follow the road back to the main Palace Gate, keeping the building on your right-hand side. This is the main public entrance to the Palace.

The ground level dimensions of this whole gate are 10 metres x 10 metres and it is 20 metres high in total. The arched empty space in the gate, through which pedestrians walk is 5 metres wide, 10 metres deep, 15 metres high to the top of the arch and 12.5 metres high to the lower level of the arch.

a. Make a 3-D sketch of this arch, clearly showing measurements.

b. Assuming that apart from the central space the rest of the gate is solid stone, what is the volume of this stone? (HINT: You may have to estimate π.)


13. The Formal Gardens – The Water Terraces

Now walk through Flagstaff Gate and follow signs to the Gardens. Make your way to the Water Terraces.
It took five years, from 1925 to 1930 for the Water Terraces to be built and involved an immense amount of thought and planning. Today, these terraces are reminiscent, on a smaller scale, of the Parterre d’Eau at Versailles.

Now it is YOUR TURN!! Devise a Mathematical question based on these gardens.

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