Question 1:
SPRING TENSION
Students in the year 11 Physics class where investigating how much different springs stretched when different masses were applied. The results of the two experiments are listed in the tables below;
|
Length of Spring 1 (in cm)
|
Force exerted by the mass applied (in Newtons)
|
|
Length of Spring 2 (in cm)
|
Force exerted by the mass applied (in Newtons)
|
|
4
|
0
|
|
7
|
0
|
|
5
|
1.5
|
|
8
|
2
|
|
8
|
6
|
|
9
|
4
|
|
10
|
9
|
|
11
|
8
|
|
12
|
12
|
|
12
|
9
|
a) What are the lengths of the un-stretched springs?
b) Plot a graph of each of the test results on the same Cartesian plane.
c) Draw a straight line through the points that best describes the data and determine a function for each straight line.
The gradients of the graphs you have drawn give an indication of the ‘stiffness' of the spring. The larger the gradient the stiffer the tension in the spring.
d) Comment on the stiffness of the two springs.
e) Is it likely that these two springs will ever be the same length at a given applied force? Use algebraic and graphical reasoning to justify your response discussing the strengths and limitations of you answer.
Question 2:
The height of a boulder thrown vertically from the top of a cliff, as shown in the graph, is given by the function:
h(t) = 40 + 20t - 5t2
where;
t is the time in seconds.
h is the height in metres.
(a) Determine the average rate of change in the height of the boulder :
(i) During the first 2 seconds of flight using a graphical method.
(ii) Between t = 3 and t = 5 using an algebraic method.
(b) On any graph which displays the height and time of a projectile the "Speed" of the projectile at any time can be calculated by determining the instantaneous rate of change in the height.
Use tangents to determine the speed of a boulder at each of the following times after it is thrown. (t = 2, t = 3, t = 4, t =5)
(c) The "Acceleration" of any object is defined as being the rate at which its speed is increasing over time. The boulder which is thrown in this case reaches its maximum height after 2 seconds and thenfalls back to earth under the influence of gravity.
Use the speeds which were calculated in (b) to determine an approximate value for the acceleration which the boulder experienced on its plummet back to the earth.
The method which is used to complete this calculation should be clearly explained and justified with consideration given to the validity of the determined value of gravity.
QUESTION 3: BREAK-EVEN ANALYSIS
Break-even analysis is a technique widely used by production management and management accountants. It is based on categorising production costs between those which are "variable" (costs that change when the production output changes) and those that are "fixed" (costs not directly related to the volume of production).
Total variable and fixed costs are compared with sales revenue in order to determine the level of sales volume, sales value or production at which the business makes neither a profit nor a loss (the "break-even point").
The Break-Even Chart
In its simplest form, the break-even chart is a graphical representation of costs at various levels of activity shown on the same chart as the variation of income (or sales, revenue) with the same variation in activity. The point at which neither profit nor loss is made is known as the "break-even point" and is represented on the chart below by the intersection of the two lines:
A manufacturer of heavy machinery (production of x units per day) estimates that his total costs (in thousands of dollars per day) are given by a cost function defined by;
T = 0.25x2 +2.5x +22, x ≥ 0.
Market research shows that he can sell an average of 3 units per day at a price of $11,400 each, 4 units per day at a price of $10,200 each. Raising production to an average of 5 units per day requires him to reduce the price to $9000 each in order to sell all that he produces. If production is raised further to 6 units per day, then a selling price of $7,800 each just allows him to sell all of them.
(a) Develop a quadratic function to model this data of production and selling price. (Show all working detailing how you arrived at your function)
(b) Use your graphics calculator (or other suitable graphing application) to produce graphs of T, R (Revenue) and Fixed costs and so find the manufacturer's daily break-even point. (Copy the graphs clearly onto your answer paper). Confirm this value algebraically.
(c) From the graphs decide how many units should be made each day to maximise company profit.
(d) Based on this production model, what are the limitations for sales? Fully justify you response based on the graphical evidence you have calculated.
QUESTION 4: PREDICT THE FLIGHT PATH
A love-struck Marshall is desperate to sing songs of serenade to his beloved Bo. Unfortunately she is fast asleep in her bedroom and his first challenge is to awaken her gently from her slumber.
After much thought he decides that the best way to achieve this is to hit a golf ball at her bedroom window so that it strikes the glass but doesn't break it.
From his knowledge of Maths and Physics he remembers that the path of the golf ball will be parabolic in shape and the window will not be broken if the instantaneous rate of change of the ball's flight is;
-1m/m rate of change 1m/m.
As he stands there contemplating his shot he estimates the following measurements about Bo's palatial residence.
He will hit the ball from the sidewalk a distance of 2 metres back from the 2 metre tall fence which surrounds the house. The distance from the other side of the fence to the front wall of the house is 10 metres.
Bo's bedroom window is 1 metre high and the bottom of the window is 3.5 metres above the ground.
(a) Determine a suitable quadratic function which will ensure that Marshall is able to fulfil his burning desire to serenade his princess whilst ensuring that the glass in the window isn't broken, which will almost certainly wake her father who it is safe to say would be far from impressed with his enthusiasm.
(b) Marshall's friend tells him there is more than one possible solution for the function describing the flight of the golf ball. Investigate this claim and if true, include the variables that have led you to this conclusion. What assumptions could be made in your investigation and support these with graphical sketches identifying how these will result in different mathematical models for the flight of the golf ball. Provide and alternative solution(s) based on this investigation.