MA 001 - VÅR 2003

Obligatory problem set  1

Each student registered for the spring 2003 exam in MA 001 must hand in acceptable solutions to two obligatory problem sets. This is the first problem set.
Each student prepares and hands in an individual solution to his or her group-teacher or to the mathematical institute,  7th floor in the  N.H. Abels building. The group-teacher then corrects and grades the solutions.  Those qualifying for a passing grade (4.0/E or better) are considered accepted.

Give a mathematical justification for all answers. Give exact answers when possible.
Remember to write your full name and the number of your group on the solution you hand in.

The absolute deadline for handing in solutions is  fredag 28. februar 2003 kl. 14.30.
Only applications with documentation from a doctor may give postponment on delivering in the solutions. Applications are sent to Matematisk Institutt v/ Heidi Raude (heidimr@math.uio.no).
 

Problem 1

Body Mass Index (BMI) is a measuring unit used to determine when a person is underweight, has average weight or is overweight.
A persons BMI is calculated by dividing the weight (measured in kilo) by the square of the height (measured in meter).
Making this into a formula, we get

B =  v/(h^2)

Where
B is Body Mass Index,
v is weight,
and h is height
A person is considered as average if 25 < B < 30,
underweight if  B < 25
and overweight if  B > 30

a) Assume that B = 25
Use the expression above to express the weight as a function of the height. Plot the graph for the expression you get into a coordinate system, choose appropriate units on the axes.

b) Assume that B = 30  and plot the weight as a function of the height in the same plot as over. On your plot, shade the area that shows the points where B is between 25 and 30.
 (I.e. the combinations of weight and height that indicates a person with average  Body Mass Index.)

Two mathematics teachers : Pelle og Tor are overweight.
Pelle is 1.80 m  and weighs  100 kg
Tor is  1.80 m and weighs 140 kg

c) How many kilo must Pelle loose to get a BMI below 30 ? (Find the solution by computation)

d) Tor thinks it will be easier to grow taller than to loose weight. How tall must Tor be to get a BMI below 30?(Find the solution by computation)
 

Problem 2

 Ranestad-Air carries passengers between town A and town B.
On each flight there is room for x passengers paying low fare, and y passengers paying full fare. A low fare ticket costs 300 NOK and a full fare ticket costs 1500 NOK.
The airline has these conditions for their flights:
  1. There is room for 150 passengers on the plane.
  2. Low fares promotes sale, so they never sell more full fare tickets than low fare tickets.
  3. To keep the budget in balance, the company needs to sell tickets for at least 60.000 NOK to each flight

    4.  
a) Write down the  5 inequalities that x and y needs to satisfy. Explain which inequality comes from which condition.

b) Shade the region in the xy-plane where x and y satisfy these 5 inequalities. Find the x- and y- coordinates of the corners of the region.

c) Make an expression describing the airline's sales as a function of x and y,  and sketch some level lines for this function in the same coordinate system as used in problem b). Find the number of low fare and full fare tickets that makes the largest sale. How large a sale is possible for one flight ?

d) How many low fare tickets can the company sell without going below budget?
 

Problem 3


A tunnell is being built between the points C and D located on different sides of a mountain.  The figure depicts distances and angles measured from two points, A and B. Use these measurements to compute the length of the planned tunnell.