Homework answers / question archive / Unit 1: Mathematical models Assessment of learning: Teacher-marked This is an assessment of learning, which is used to evaluate your work based on established criteria and to assign a mark

Unit 1: Mathematical models Assessment of learning: Teacher-marked This is an assessment of learning, which is used to evaluate your work based on established criteria and to assign a mark

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Unit 1: Mathematical models

Assessment of learning: Teacher-marked

This is an assessment of learning, which is used to evaluate your work based on established criteria and to assign a mark. Your teacher will provide you with feedback and a mark. This

Assessment of Learning is worth 17.5% of your final mark for the course.

Instructions

There are 4 Tasks in this assessment of learning.

Task 1: Knowledge and Understanding (25 marks)

Show all work for each question.

1. Evaluate.

a) 27^1/3

b) 32^3/5

c) 8^5/3

d) 25^-1/2

e) (64/27)^-1/3

f) 250^1/4

g) 1.28^3.7

h) (3/5)^-5

i) ⁶Ö28

2. Simplify.

a) (a³)²

b) n⁵/n³

c) (3x²)³

d) x(x²)(x⁴)

e) t²(t⁵)/(t³)²

3. Solve 4^x = 36, accurate to three decimal places.

4. Solve.

a) 2^x = 64

b) 3ⁿ⁺⁴ = 27²ⁿ

c) 4^2(x+5) – 11 = 245

Task 2: Applications

Show all work for each question.

5. A rock sample containing 256 g of tungsten-187 decays unit it contains only 0.25 g of tungsten-187. If the tungsten has a half-life of one day, how long would this take?

6. An investment can be modelled by the equation A = 2500(1.06)^t, where A represents the final amount of the investment and t represents the time in years. How many years would it take to have a final amount equal to $10 000? Round your answer to the nearest tenth of a year.

7. The surface area of a cylinder is given by the formula SA = 2πr² + 2πrh, where r represents the radius and h represents the height. Solve the formula for h and then use it to complete the following table accurate to two decimal places.

	Cylinder A	Cylinder B	Cylinder C
Surface Area, SA (cm2)	2500	5000	4275
Radius, r(cm)	8	6	15
Height, h(cm)

 

8. Kinetic energy refers to the energy of an object in motion. The amount of energy an object has (K) is related to the mass of the object (m, measured in kg), and its velocity (v, measured in m/s). The formula used to calculate kinetic energy is K = ½ mv².

a) Solve for K if m = 20 kg and v = 18 m/s.

b) Solve for m if K = 486 and v = 9 m/s.

c) Solve for v if K = 345 and m = 7 kg.

Task 3: Communication

Show all work for each question.

9. Identify what type of equation (linear, quadratic or exponential) was solved in parts b) and c) of the previous question (question #8). Justify your choice.

10. A liquid is heated in a lab using a hot plate. The graph of temperature vs. time is as follows.

Answer the following questions using the letters A through D on the graph above.  More than one letter may be required for each question. (5 marks)

a)  Where is the rate of change zero?

b)  Where is the rate of change changing?

c)  Where is the rate of change positive?

d)  Where is the rate of change greatest?

11. The discus throw is a sport in which competitors try to throw a metal disk as far as possible. Use the graph showing the winning discus throws from the Olympics to answer the questions that follow.

a)  What general trend can be described from this graph?

b)  Does the relationship appear to be linear?  Explain.

c)  Give reasons that may explain the shape of the graph.

d)  Use the graph to predict the winning distance for the discus throw inthe 2020 Olympics.

12. Two sisters, Bethany and Dana, decide to invest for their retirement. They both expect to retire in 40 years. The following graph shows their investment.

a)  Describe the rates of change for each person using words like increasing, constant, or zero rate of change. 

b)  The following phrases can be used to describe the sisters investments.  Use all of the

phrases by placing the phrase under the correct sister’s name in the following table.  Use

each phrase only once to describe the sisters’ investment.  Include reasons for matching the

phrase to that sister. (4 marks)

  Had $2500 to invest.

  Invested at 8% compound interest.

  Invested at 8% simple interest.

  After 20 years the investment improved to 11% simple interest.

  Stored her money under the mattress for two years, then invested it.

  Had $5000 to invest.

Bethany	Dana

 

 Task 4: Thinking (25 marks)

Show all work for each question.

13. The company you work for has downsized and you have lost your job. You receive a severance package of $65 000 and decide to invest it for retirement, earning an average of 7% per year. It has been suggested that you should be able to retire comfortably with $500 000 in savings. If your investment can be modelled with the equation y = 65000(1.07)^x how many years will pass before you reach $500000?

14. A large piece of aluminum is gradually pressed into a thin sheet by passing through rollers. Each pass through the rollers results in a thinner piece of aluminium according to the equation A = 80 (0.75)^x, where A represents the thickness of the aluminum (in millimetres), and x represents the number of passes through the rollers. How many passes are needed to end up with a 1.5 mm thick sheet of aluminum?

15. The following two cylinders of equal size are to be constructed so that the combined total volume is 20 m³.

If the volume of a cylinder is V = πr²h, determine the height if the radius is to be 1.2 m.

16. Two investors, Brent and Lincoln, invest different amounts of money at the same time for an 8-year period. Brent invests $10 000 at 6% per year and his final amount is given by the formula A = 10000(1.06)⁸. Lincoln is shrewder, and was able to invest only $8000 and end up with the same amount at the end. His final amount can be modelled using the equation A = 8000(1+ i). Determine the value of i for Lincoln and the yearly interest rate his money earned to allow this to happen.

17. In many cases, a worker’s hourly wage depends on the number of hours worked. If a certain number of hours is reached, the pay rate beyond that point increases. When working beyond a certain number of hours, some employees earn time-and-a-half, or 1.5 times their regular pay, for the remaining hours. This type of arrangement is shown in the following graph of pay vs time.

a)  At what point does the rate of change increase?  (1 mark)

b)  Calculate the rate of change on either side of the point from part a). Show your work and include units.  (2 marks)

c)  Use your answers from parts a) and b) to describe this worker’s pay structure in terms of  hourly pay, and when the hourly pay changes.

Feedback and marking You will receive three kinds of feedback: Your teacher will highlight the phrases on the rubric that best describe your assignment to  show you how you have done.  Your teacher will also provide you with detailed comments about the strengths of your  assignment, the areas of the assignment that need improvement, and the steps you should  take before submitting another assignment like this one.  The final piece of feedback that you will receive will be your mark. Each of the four categories of knowledge and skills is weighted equally at 25 marks, making a total of 100  marks. The final mark on this Assessment of Learning is determined by your teacher based on their professional judgement of the requirements for the assignment.