What is the Gradient of the Line Described by This (y = -½x+3) Equation?

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What is the maximum profit that can be made by the car company?

 

SKU: Math784222

1. a) Consider the equation y = -½x+3

What is the gradient of the line described by this equation?

What is the y-intercept of this equation?

 

b) Find the equation of the line that has a gradient of 5 and passes through the point (-5,-2).

c) Find the equation of the line that is perpendicular to the line in part b) and passes through the point (2, 1)

Your answer should be left in fractional form.

 

2. a) Solve the equations below simultaneously using the graphical method:

(Hint: Draw up a table of coordinates for each equation and plot on the same set of axes. Use x values between -4 and 4)

y=-x²+1

y=x-5

graph

Indicate the location of the solution(s) on your graph and clearly state the solution(s) in the space below:

 

3. The following graph shows the World Record Times for the 200 m race for flying start machines (a kind of bicycle).

graph.1

a) How would you describe the correlation between Year and World Record Times

b) Estimate the correlation coefficient.

c) Draw a line of best fit by sight on the graph above.

d) Find the equation of the line of best fit.

e) Use your equation to predict the World Record Time in 1970.

f) Comment on the reliability of your answer in part d).

 

4. a) Factorise the following:

i) y=x²-x-42

ii) y=6x²+11x-10

b) Solve the quadratic equation in part a) (ii) using a method of your choice.

 

5. Some chickens are kept in a square enclosure with sides measuring x metres.

If the number of chickens is increased, the enclosure will have to be made bigger. The enclosure will have 1 metre added to one side and 2 metres added to the adjacent side.

a) (i) Draw a labelled diagram of the original enclosure in the space below.

    (ii) Add to the diagram to show the new enclosure. Mark the lengths of each side on your diagram.

b) Find a simplified expression (i.e. no brackets) for the area of the new enclosure

c) If the new enclosure has an area of 42 square metres, what were the dimensions of the original enclosure?

 

6. The profit per car, $P, made when a particular model is manufactured in t hours is given by the equation:

P=-4t²+200t-1500

a) Sketch a graph showing the relationship between profit (y-axis) and time of manufacture (x-axis).

Show the coordinates of the y-intercept only.

b) How long does the manufacturing process need to take in order for the company to make the maximum profit?

c) What is the maximum profit that can be made by the car company?

d) Indicate the position of the turning point on the sketch graph in part (a) and write down the coordinates of this point in the space below.

 

7. The size of a colony of rare termites is related to the size of the colony’s mound. The relationship can be modelled using the equation below:

h = −N²+ 25N − 5

Where N is thousands of termites and h is the height of the mound in centimetres.

a) Sketch the graph showing the relationship between the number of termites (x- axis) and the height of the mound (y-axis).

Show the coordinates of the y-intercept only.

b) How many thousands of termites are required to build a mound of maximum height?

c) What is the maximum height of a mound predicted by the model?

d) Indicate the position of the turning point on the sketch graph in part (a) and write down the coordinates of this point in the space below.

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