CST Math 2015 - Day 10 - Situational Problems [PDF]

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C1 – Situational Problems Secondary IV: Cultural, Social and Technical Mathematics

+ C1- Situational Problems 

Highlight the end question.

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Divide the question up into steps or sections that you will solve 

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Determine what concept is being covered in each step.

Go back to the original question to make sure that your final answer answers the question.

The Vegetable Field – June 2013

+Let’s Make a Plan 

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We have to find the two possible measurements of the angles in the squash field. But when we look at it, those measurements are in direct relation to the area of the field, which is linked to the cost of the field. In order to see if our cost for the squash field is accurate we have to know what the cost of the other 4 fields will cost.

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Step 1: The Carrot Field 

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Step 3: The Lettuce Field 

Key word “percentile”

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Find the 72nd percentile of 12 terms

Step 4: The Squash Field 

Key words “increased by 5% each year” this tells us that we have an exponential function 

Write the rule

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Solve for F(x)

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Step 2: The Potato and Broccoli Fields 

First we have to determine the length of line QR Next we have to determine two possible measurements for the missing angles 

(this is an educated guess; 180 – 100 = 80 therefore two angles that add up to 80 that are somewhat close in size

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Next use Sin Law to determine another side of the triangle

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Next find the area of the triangle using Sin Area

In order to find the cost we need to know the area of each field 

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The area if the potato field 

This is a right triangle therefore we can use the basic triangle formula to find the area

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First though we need to find the height (line AD) - Metric Relation

The area of the Broccoli field 

Similar triangles – Therefore we can use the rule of proportionality to find all the side lengths

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Hero’s formula to determine the area.

Once we have the areas of the two triangles we need to find the cost of each field. 

Using the chart we can find the systems of equations to determine the two variables

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Then multiple the cost per meter of each vegetable times the area of the field to get your total cost.

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Distance formula

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Once you have the area look at the two rules in the piecewise function and determine which rule does the area satisfy

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Solve for C(a)

Step 5: Add all the cost of each field and make sure that they fall within the budget .

+Step 1: The Carrot Field  

The   farmer plants carrots every year in the same field. In 2010,the cost to grow the carrots was $2000 and has increased by 5% every year. The farmer would like to know how much it will cost to grow carrots in 2013.

Exponential Function: f(x) = acx Cost

Initial Value

f(x) = 2000(1.05)3 f(x) = $ 2315.25

Rate

c = 5% = = 1.05 Time in years

+Step 2: The Potato + Broccoli Fields 

The Area of the Potato Field   

The farmer plants poatoes in a field shaped as traingle ABC shown below. The length of is 9 m and the length of is 16 m 

To find the area we need the height 

Metric Relation 

Label the diagram to match your memory aid

=dxe = 9 x 16 f = 12 

Find the area of the triangle ABC

• Area = • Area = • Area = 150

+Area of the Broccoli Field  

The   farmer plants broccoli in a field shaped as triangle SVW shown below. The length of is 5.1 m, the length of is 9.75 m and the length of is 12.75 m. Triangle STU is similar to triangle SVW. 

Because we have similar triangles we can find the length of &

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We can use proportionality to determine the side lengths 

= =

15.3(SW) = 260.1 SW = 17 m

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= =

15.3(WV) = 198.9 WV = 13 m

Area of the triangle – Hero’s Formula   • Area = • •

• • • •

P P = 25.2

Area Area Area Area

= = = = 110 m2

+Finding the cost of the Potato and Broccoli Field 

The farmer does not know the cost to grow the potatoes and broccoli. Luckily, two neighbouring farms have also grown potatoes and broccoli. The cost per m2 to grow the potatoes is the same for all of the farms. The cost per m2 to grow the broccoli is the same for all of the farms.

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We have two unknown’s (cost of the potatoes and cost of the broccoli), therefore we have a systems of equations 

Write the two rule

In order to solve we have to decide of 1 of the 3 methods (it doesn’t matter which one) Substitution method – First isolate the y variable in one equation

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80x + 320y = 7600 60x + 120y = 3300

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60x + 120y =

Substitute in Y 120y = -60x + 3300 80x + 320(-0.5x + 27.5) = 7600 3300 y = -0.5 + 27.5 80x – 160x + 8800 = 7600 -80x = -1200 x = 15$ per m2 Solve for y 80(15) + 320y = 7600 1200 + 320y = 7600 Cost to grow Potatoes = Area x Cost per m 2 320y = 6400 = 150 m2 x $15 2 Y = 20$ per m = $2250 Cost to grow Broccoli = 110 m2 X $20 = $2200 Total cost = $4450

+Step 3: The Cost of the Lettuce Field  

The   following is a list of the growing costs of 12 of the lettuce fields in the region. The farmer’s lettuce occupies the 72th percentile within the list.

$440, $460, $465, $485, $495, $495, $500, $515, $520, $525, $530, $540 

What place is the 72th percentile: 

Position = X # of data values

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Position = x 12

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Position = 8th number = $515

+ Step 4: The Squash Field  The   farmer plants squash in a field shaped as triangle PQR in the diagram 

below. The coordinates of points Q & R are given and the scale is in metres. The measures of angle QPR is 100. The measures of angles PQR and PRQ are unknown.

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Remember that this is the main part of the question that we are trying to solve. What is one possible set of measures for angles PQR and PRQ in the squash field We know that the sum of the interior angles of a triangle is 180, therefore the sum of the two remaining angles is 80 The farmer is concerned that the squash won’t grow properly in the corners of the field. He must ensure that all of the angles in the corners of the squash field measure at least 30 The two other angles look somewhat close in measure, one being slightly bigger then the other, so let’s say that