maths

Answer all of the following questions. You may use any resources that are available, but your work must demonstrate your logic clearly. Working with partners are encouraged but keep in mind that none of the works can be identical. You may consult your teacher if you need any assistance. Final deadline for the submission is Monday, April 16th, 2018. Any submission beyond this date will be subject to 5% deduction per school day.

1. Reference: Unit 1 Polynomial and Rational Functions

[Source: Advanced Functions, Nelson]

A New School

Researchers at a school board have developed models to predict population changes in the three areas they service. The models are A(t)=360t+6;????(????)=30????????+1;????(????)=5041?2???? for areas A, B and C respectively. For all the functions, the population is measured in thousands and t is the time, in years, since 2007. The existing schools are full, and the board has agreed that a new school should be built. Based on the given information, in which area should the new school be built, and when will the new school be needed?

Perform the following steps to aid your investigation.

(a) Graph each population function for the 20 years following 2007. Use your graphs to describe the population trends in each area between 2007 and 2027.

(b) Describe the intervals of increase or decrease for each function.

(d) Determine the intervals over which

* The population of area A is greater than the population of area B

* The population of area A is greater than the population of area C

* The population of area B is greater than the population of area C

(e) Determine when the population of area B will be increasing most rapidly and when the population of area C will be increasing most rapidly.

(f) What will happen to the population in each area over time?

(g) Decide where and when the school should be built. Compile your results into a recommendation letter to the school board.

2. Reference: Unit 2 Trigonometry

Choose and prove the following trigonometric identities.

(a) cos????+sin????cos?????sin????+cos?????sin????cos????+sin????=2sec2????

(b) 12[tan(????+3????4)+tan(????+????4)]=tan2????

3. Reference: Unit 3 Exponential and Logarithmic Functions

Acidity of a solution refers to the level of available hydronium ion concentration. Higher the concentration of hydronium ions means there are more ions available to participate in a reaction readily, which makes the solution more reactive. For instance, concentrated hydrochloric acid is very corrosive because of the high concentration of the hydronium ion.

(a) State the formula of the pH scale and briefly explain how it indicates the hydronium ion concentration.

(b) How many folds difference of the hydronium ion concentration do you expect to see between two solutions with pH 3 and 4? Using this information, explain the implication of the difference of 1 pH scale.

(c) You are supposed to prepare a solution with pH of 4.5. Given that you have 10 ml of stock solution of hydrochloric acid with pH 0.2, how much of water is needed to prepare 500 ml of the solution with pH 4.5?