Matematik B. Højere handelseksamen. Mathematics Level B. Thursday August 17, hhx172-mat/b

Transcription

1 Matematik B Mathematics Level B Højere handelseksamen hhx172-mat/b Thursday August 17,

2 Mathematics B The paper consists of two partial tests. The partial test without Tools consists of assignment 1 to 5, i.e. all in all 5 questions. The answer to this partial test must be handed in by 10am. The partial test with Tools consists of assignments 6 to 10, i.e. all in all 13 questions. The 18 questions each carry a weight of 5 points in the assessment of the total paper. You are only to hand in one of the assignments 10A, 10B and 10C. If you hand in more of these assignments, only the first assignment will be assessed. During the first hour of the test tools apart from writing and drawing tools are not allowed. In the last 3 hours of the test all tools are allowed. In the assessment of the answers to the individual question and the overall impression emphasis will be put on whether the examinee s train of thought is clearly reflected in the answers. The paper must account for the method used and documentation in the form of a sufficient number of intermediate results and/or a mathematical explanation of the use of different facilities offered by an IT program. If graphs and illustrations are used there must be a clear relationship between text and illustration. The following three data files belong to this set of exam questions: Wheat Lunch Bar

3 Hhx Mathematics B August 2017 page 1 of 7 The partial test without tools At Exercise 1 During a period of 10 days a crossfit training center has registered the number of people signed up for the afternoon training. The number of people signed up spread out as follows: a) Find the average number of people signed up. Exercise 2 a) Draw the graph of a function f fulfilling the following: The range R: y 6;8 The function has more than one zero/root The function is increasing on the interval [3 ;7] The point ( 2, 4) sits on the graf Attachment 1 may be used. Exercise 3 a) Account for x 7 being a solution to the equation x 3 6 x 3. Exercise 4 In a country the number of dollar millionaires increases by 4.5 % per year, and in year 2016 there were 45,600 dollar millionaires. a) State an exponential model describing the development of the number of dollar millionaires after year 2016.

4 Hhx Mathematics B August 2017 page 2 of 7 Exercise 5 The correlation between the demand for a commodity (in tons) and the price (per ton in DDK) is found by the function dx ( ) 0.15x 300, 0 x 1500 x denotes the quantity demanded and d (x) denotes the price. The correlation between supply of a commodity (in tons) and price (per ton in DKK) is found by the function sx ( ) 0.10x 50, 0 x 1500 x denotes the supplied quantity and s(x) denotes the price. The equilibrium quantity is defined as the quantity of the commodity making supply and demand equal. a) Find the equilibrium quantity of the commodity. price in in DKK d d s 200 s quantity (in tons) quantity (i tons) The answers to the test without tools must be turned in by 10.00

5 Hhx Mathematics B August 2017 page 3 of 7 The partial test with tools At Exercise 6 a) The equation 2e x 3 13 has been solved below. Explanations to the calculations must be given. Attachment 2 may be used. 2e x 3 13 The equation has been stated. 2e x 10 e x 5 x Exercise 7 In the company SYNCORF the cost of the production of a given chemical can be described by the function Cx x x x x 3 2 ( ) , 0 C(x) is the total cost per week at a production quantity of x liters of the chemical. The company can sell the whole production at DKK600 per liter. The revenue can therefore be described by the function Rx ( ) 600 x, x 0 a) Find the production quantities of the chemical making cost and revenue equally big. The profit P can be found by profit = revenue cost b) Find the profit at a production quantity of 150 liters. c) Find the maximum profit the company can obtain per week by the production of the chemical.

6 Hhx Mathematics B August 2017 page 4 of 7 Exercise 8 The yield from a filed of wheat depends on many factors such as the weather and growing conditions. The yield in Denmark is approximately normally distributed with a mean of 8.1 tons per hectares and a standard deviation of 1.4 tons per hectares. a) What is the probability of obtaining at least 10 tons per hectare on a field chosen at random? b) Find the 10 th percentile of the yield and account for the significance of this number. An association of smaller farms wants to examine whether these farms give the same yield and they make a sample with answers from 42 smaller farms. The table shows an excerpt of the results of the sample given in the file wheat. Yield (tons per hectare) : c) Find the mean and the spread of the yield of the small farms. d) Find a 95% confidence interval for the mean and assess whether the small farms get the same average yield per hectare as the whole country.

7 Hhx Mathematics B August 2017 page 5 of 7 Exercise 9 A company Gabasound Ltd. produces two different kinds of speakers, GIGABOOM and MEGABOOST, which they sell to electronics chains. The production of the speakers goes through three processes. At first the speakers are assembled, then they are tested and at last they are packed. In the assembly department 1 hour is used per GIGABOOM and 4 hours per MEGABOOST. There are a total of 1600 hours available per week in the assembly department. In the test department 10 minutes are used per GIGABOOM and 10 minutes per MEGABOOST. There are a total of 5500 minutes available per week in the test department. In the packaging department 6 minutes are used per GIGABOOM and 4 minutes per MEGABOOST. There are a total of 3000 minutes available per week in the packaging department. In the present market GIGABOOM is sold at a profit of DKK 100 per unit, whereas MEGABOOST is sold at a profit of DKK 200 per unit. Let x denote the number of produced GIGABOOM per week and y denote the number of produced MEGABOOST per week. The function f(, xy) a x b ydenotes the total weekly profit. a) Find an equation for the function f. b) Find the number of units of the models GIGABOOM and MEGABOOST the company must produce to obtain the biggest possible weekly profit. The production found in sub-question b) means that there is a surplus amount of time in one of the departments. c) Find the surplus amount of time in the given department.

8 Hhx Mathematics B August 2017 page 6 of 7 You may only turn in one of the exercises 10A, 10 B, and 10C for assessment. If you turn in more of these exercises, only the first exercise will be assessed. Exercise 10A In a survey from 2016 made by Kompas Kommunikation for 3-stjernet cold cuts 1340 representatively chosen Danes were asked what they eat for lunch most of the time. The table below shows an excerpt of the data found in the file Lunch. Preferred lunch Gender A Open-Faced Sandwich Woman B Sandwich Man C Salad Woman : : a) Make a table like the one below containing data from the survey. Woman Man Total A Open-Faced Sandwich B Sandwich C Salad D Pasta E Hot Dish F Doesn t eat lunch G Other options Total stjernet wishes to examine if there is a correlation between the Danes answers to the question and their gender. b) State a hypothesis that can be used to test this correlation and test the hypothesis at a significance level of 5%. Source: 3-stjernet

9 Hhx Mathematics B August 2017 page 7 of 7 Exercise 10B A building society offers an annuity loan on the following conditions: The principal DKK Maturity 120 months Monthly interest rate 0.33% Monthly payment DKK A customer wishes a lower monthly payment of DKK The rest of the conditions stay unchanged. a) Find the maturity of the customer s loan. b) Find the size of the last payment. Exercise 10C A bar has made a survey of the age of 105 customers and the amount the customer spent visiting the bar. The table below shows an excerpt of the data found in the file Bar. Age Amount a) Make a scatter plot of the correlation between age x and amount y and make a linear regression model Bx ( ) a x b, describing this correlation. b) Using the model, find the age of a customer using DKK 250 at her visit to the bar.

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