Determining minimum hiking time using DEM

**Order Instructions:**

A supermarket has the following two offers on a particular brand of mature cheddar cheese.

Offer (1) 30% off the usual price Offer (2) 30% extra free

The usual price for a 350 g pack of this cheese is £4.50, which works out at £1.29 per 100 g.

Calculate the price of a 350 g pack of this cheese under offer (1).

(ii) Calculate the price per 100 g of this cheese under offer (1).

Calculate the weight of a pack of this cheese if it costs £4.50 under offer (2).

Calculate the price per 100 g of this cheese under offer (2)

A rival supermarket introduces the following offer on a 350 g pack of this cheese that also costs £4.50.

Offer (3) Buy one, get another one half price

Find the price per 100 g of this cheese under offer (3).

Calculate the percentage decrease in price for this cheese under offer (3) compared to the usual price.

Compare these three offers from a shopper’s perspective. In your answer, you should summarise the above information by copying and completing the table below, and then consider both the price and the quantity to be bought to obtain the offers.

Offer 1

Offer 2

Offer 3

Quantity (in g)

Total cost (in £)

Price/100 g (in £)

Question 2

At the weekend, Callum walked 8 km on Dartmoor. He used a map with scale 1 : 25 000 to determine distances.

(a) On the map, the distance that Callum walked before stopping for lunch is 18.4 cm. Find the corresponding distance on the ground in km.

(b) (i) The complete walk of 8 km took Callum 2 hours and 15 minutes. Find his average speed in km/h.

(ii) Convert your answer from part (b)(i) into metres per second. Give your answer correct to two significant figures. [3]

(c) The map shows that the walk started at a height of 180 m above sea level, then dropped 60 m, climbed 250 m and finally dropped 150 m again. Let h represent the height above sea level in metres.

(i) Find the highest and lowest heights above sea level encountered on the walk.

(ii) Draw a number line to represent the interval of the heights above sea level encountered on the walk.

(iii) Use a double inequality to show this range of heights above sea level encountered on the walk.

(d) Callum has adapted Naismith’s Rule to estimate the time that he takes to walk up a scenic hill taking into account pauses to look at the views. The time is given by

T=_D__+_H___+_D___

5 600 25

where

T is the time for the walk in hours,

D is the horizontal distance walked in kilometres, H is the height climbed in metres.

On one section of his walk, Callum climbed 140 m over a horizontal distance of 2.6 km, with scenic views. Find the estimated time (to the nearest minute) that Callum’s formula gives for him to climb this hill.

QUESTION 3

(a) The following formula (known as the Carroll/Huntington Formula) is commonly used to estimate W, the body-weight of a horse in kg:

W= ____g2l_______

11800

where

g is the girth of the horse in cm,

l is the length of the horse in cm.

Estimate Quin’s body-weight, to the nearest 10 kg, when his girth measurement is 189 cm and his length is 161 cm.

In order to lose weight, each day an overweight horse should eat hay weighing 1.5% of its ideal body-weight.

(i) Using this information, copy and complete the table below.

ideal body weight of horse (I in kg) 200,300,400,500, 600,700,800

weight of hay (H in kg) 3 4.5 6

Draw a graph to illustrate this information. The vertical axis should show H, the mass of hay in kg, and the horizontal axis should show I, the ideal body-weight of the horse in kg. Mark the points clearly, and join them up with a line. You will find the tips for drawing graphs on page 84 of Unit 2 useful.

You can draw your graph either by hand (using graph paper) or using a computer.

(iii) Quin is overweight and being fed 6.5 kg of hay each day. Explain how you could use your graph to find his ideal body-weight to the nearest 10 kg.

(c) The farrier charges £65 to shoe a horse, £20 to trim an unshod horse’s hooves, and £10 for travelling.

(i) Ursula has two unshod horses and three shod horses. Calculate how much the farrier will charge on a visit to shoe the shod horses and trim the hooves of the unshod horses.

(ii) Construct a formula for the cost C, in £, for a visit by the farrier to shoe s shod horses and trim the hooves of t unshod horses.

**SAMPLE ANSWER**

**Question 1**

Prices under offer (1)

Therefore, the price is £4.50 – £1.35 = £3.15

Therefore, the price is £1.29 – £0.387 = £0.903

Prices under offer (2)

- If 70% = 450 g

100% =?

- 100% = £1.29

70%

Prices under offer (3)

Buy one, get another one half price

One 450g pack = £4.50

Half of £4.50 = £2.25

So, two 450g packs cost = £4.50 + £2.25 = £6.75

Therefore,

If 900g = £6.75

Then, 100g =

Percentage decrease

£1.29 – £0.75 = £0.54

Comparison table

Offer 1 | Offer 2 | Offer 3 | |

Quantity (in g) | 540 | 742.857 | 1000 |

Total cost (in £) | 5.403 | 5.403 | 7.29 |

Price/100g (in £) | 0.903 | 0.903 | 0.75 |

**Question 2**

- 1 : 25 000

This means that 1 cm on the map represents 25 000 cm on the ground

Therefore, 18.4 cm represents:

18.4 cm x 25 000 cm = 460000 cm

This is equivalent to 460,000cm divided by 100,000cm = 4.6 km

- (i)

Distance = 8 km

Time = 2.5 hours

(ii) 1 km/h = 0.277778 m/s

Therefore, 0.277778 x 3.2 = 0.89 m/s

- (i) 180m – 60m = 120m

120m + 250m = 370m

370m – 150m = 220m

Highest height = 370m

Lowest height = 120m

(ii) Number line

(iii)

- Naismith’s Rule

H = 140 m

D = 2.6 km or 2600 m

T = 520 + 0.23 + 104 = 624.23 minutes

**Question 3**

- Hay eating table of an overweight horse

Ideal body weight of horse (in Kg) | 200 | 300 | 400 | 500 | 600 | 700 | 800 |

Weight of hay (H in Kg) | 3 | 4.5 | 6 | 7.5 | 9 | 10.5 | 12 |

- Graph
- First is to pinpoint where 6.5 kg of hay is on the vertical axis and draw a straight line to join the line of the plot and then check the value of ideal body-weight of the horse of the horizontal axis.

- (i) Charges

£65 = to shoe a horse

£20 = to trim an unshod horse hooves

£10 = travelling

Total charge = (2 x £20) + (3 x £65) + £10

Overall total = £40 + £195 + £10 = £245

(ii) Assume x = the number of horses to shoe

y = the number to trim an unshod hooves

C = 65 x + 20 y + 10

**Reference**

Magyari-Sáska, Z. & Dombay, Ş. (2012). Determining minimum hiking time using DEM. *Geographia Napocensis* (Academia Romana – Filiala Cluj Colectivul de Geografie). Anul VI (2): 124–9. Retrieved 5 November 2014.

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