**Assignment Task**

**Task**

**Question 1. The Leaf Data.**

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Get Help Now!**1. The Leaf Length.**

**(a) **In Table 2 above, using the column labelled ‘Leaf Length – Ranked in Order’, arrange the leaves in rank order of leaf length from shortest to longest.

**(b) **Use your rank order arrangement to find the median value of leaf length within the sample. In the spare column of Table 2, draw a single arrow that points exactly to the median value.

**(c)** Describe what this median value – of the length of leaves in the sample – represents. [1]

**(d) **Calculate the mean value of the length of leaves in the sample, correct to one decimal place.

**(e)** Describe what the mean value of the leaf length represents.

**(f)** State the modal value/s of the leaf length (from Table 2) and explain what is meant by any modal samples in this context.

**2. The Leaf Ratio.**

**(a) **In Table 1, the column for Length to Width Ratio is missing one entry. Calculate the value of the missing entry.

**(b) **Regarding Length to Width Ratio, and considering the leaves with a ratio of less than 1, describe the relevant physical characteristics of these leaves.

**(c) **Regarding Length to Width Ratio, and considering the leaves with a ratio that equals 1, describe the relevant physical characteristics of these leaves.

**(d) **Regarding Length to Width Ratio, and considering the leaves with a ratio of more than 1, describe the relevant physical characteristics of these leaves.

**(e) **Calculate the length to width ratio of a leaf of length 84 mm and width 21 mm. Does this leaf “fit” with the sample of 25 leaves that you were given? Provide a practical, real-world, possible explanation behind the answer to this question.

**3. The Report.**

As part of the report on the study of the relationship between length and width in the sample of 25 leaves, complete the following sentences, filling-in the blanks by referring back to Table 1 and Table 2.

**(a) **The range of lengths of the sample of 25 leaves is the difference between the longest leaf and the shortest leaf. The longest leaf is ……….…………. and the shortest leaf is ….………………. . The range is the difference between the two and this is ……………..….. .

**(b)** The average (mean) length of leaf is …………..…… . The middle (median) length is ………….….. .The modal values of the sample of leave are……….………………………

**(c) **Approximately half (50%) of the sample is shorter than …………………. . Approximately ……………. % of the sample is longer than 78 mm. Approximately 67% of the sample leaves are shorter than ………………… .

**(d) **Determine the percentages to complete the sentence below, correct to one decimal place. The percentage of the leaf sample that has a leaf ratio less than 1 is ………..…….., while the percentage of the leaf sample which has leaf ratio greater than 1 is ………..……. . …………… % of the sample has leaf length equal to leaf width.

**4. Plotting Points.**

**(a)** Plot the Samples Numbers 1, 10 and 24 (from Table 1) as points on the axes given below. Label each of the plotted points with its sample number. Let width, w, be the explanatory variable that is plotted on the X axis (the horizontal), and let length, L, be the response variable that is plotted on the Y axis (the vertical).

The actual scatterplot, with all 25 of the Samples accurately plotted, is presented below. You should clearly see that the diagonal line drawn on the scatterplot is not the line of best fit for the sample points.

**(b)** Use a ruler to draw in your line of best fit for all the data points, and then label it as “line of best fit”.

**(c) **For all the data points, the association between length and width variables is evidently linear.Describe the strength and the direction of the association between leaf width and leaf length.

**5. Prediction. **

The linear equation that can be used for the purposes of predicting either the length (L) or the width (w) of any leaf of the ivy plant is: ???? = 1.58 ???? ? 26

**(a)** Use the equation to predict the Length of a leaf if its width is 110 mm (to the nearest mm). State whether this prediction is reasonable, and explain your reasoning.

**(b) **Use the equation to predict the width of a leaf if its Length is 75 mm (to the nearest mm). State whether this prediction reasonable, and explain your reasoning.

**2.** Choosing the right sized water tank for your home.

Rainfall is measured in units of length. It is quoted in millimetres falling on any flat, unobstructed area within the region receiving the rainfall. The amount of rainfall is recorded by a rain gauge as the height in millimetres. The Bureau of Meteorology publishes the amount of rainfall in all major population centres as an accrual over 24 hours from 9:00 to 9:00 each morning. Searching the website may lead you to a table showing the amount of rainfall every 10 minutes. A rain gauge at home can be fun for children, measuring and recording the data. Below is a hyperlink to a fact-sheet on choosing the appropriate water storage capacity for your home. Even if renting, you can perhaps arrange to install a small to medium sized water tank that could supplement your use of town water. Read the fact sheet on choosing the right sized water tank; then answer the following questions.

**(a) **The fact-sheet claims that 1mm of rain, falling on an area of 1 m2 , will deliver 1 L of rainwater into your water tank. Use your mathematical skill to show that this claim, as presented in the fact-sheet, is correct. Helpful information: 1 L is 1000 cm3 This calculation will work best using centimetres.

**(b) **A modern, medium-sized home may have a rooftop water collection area of 200 ????2 , and should get by with a rainwater tank of between 2000 and 10 000 L. Suppose that, to go with the 200 ????2

roof of your house, you would like a rainwater storage capacity of 5000 L. Calculate the rooftop rainfall (in mm, as shown by a rain gauge) that would fill a 5000 L water tank. [Note that 1 m3 is 1000 L.]

**(c)** You select a narrow tank that fits beside the wall of your house. The top (or plan view) of the water tank is a flat, composite shape with two semi-circular ends and a long straight section in-between. Thus the tank is 300 mm wide, and altogether 3500 mm long, as shown:

**(i) **Calculate the perimeter of the top of the tank (in millimetres and correct to the nearest mm).

**(ii) **You realise that the top of the tank could be useful to line up egg-laying boxes for your chickens. Calculate the area of the top of the tank (in ????2 and correct to two decimal places).

**(d) **You purchase three of these water tanks, which are 1.5 m in height. Based on your answer to (c)(ii), calculate the capacity of a single tank, to the nearest litre. Calculate whether your three new tanks, altogether, provide more than or less than your desired storage capacity of 5000 L. Then state the calculated difference (in a worded sentence).

**(e)** Once your three water tanks have been installed, the dry spell finally ends in your neighbourhood.

Steady rainfall begins to deliver 15 litres per minute from the 200 ????2

rooftop to your water tanks. Based on your answer to (d), calculate for how much time the steady rainfall must continue in order that your three water tanks reach full capacity. Give your answer in hours and minutes.

**(f) **Over time, the shape of water tanks has evolved to accommodate smaller house blocks, and more compact installation. Suppose the manufacturer still sells a different water tank whose plan view is circular. However, its walls are made from the same quantity of material that the walls of each one of your tanks is made of. Both shapes of water tank are the same height. The circumference of that circular-style tank must therefore equal the perimeter of each one of your tanks.

**(i) **Calculate the area of the top of the circular-style tank (in ????2 and correct to two decimal places).

**(ii)** Your neighbour, whose house is attached to a small farm, bought three of the circular-style tanks. Based on your answer to (f)(i), calculate your neighbour’s total water storage capacity, in litres.

**(iii) **Assuming water tanks are transported in an upright orientation, three of the circular-style tanks fit snugly in a row in the manufacturer’s truck. If a shipment involved only the narrow water tanks of the same kind that you bought, how many (at most) could fit in the same rectangular truck space? Using at least one diagram, show and/or explain your reasoning.

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