Indirect variation

1. The maximum weight (W kg) that can be supported by a piece of pine is inversely proportional to the length (L metres) of the piece of pine.

A 3 metre long piece of pine will support 250 kg.

Calculate the maximum weight that can be supported by a 5 metre long piece of this pine.

Answer.Max weight is 150 kg.

2. The number of people (N) who attend a show varies inversely with the cost per person in dollars ($C).

A show will attact 3,000 people if the admission change is $30.

(i) How many people could be expected to attend if the admission charge is reduced to $25?

(ii) What should be the cost per person if the organisers wish to attract 4,000 people?

Answer.(i)3,500 people
(ii) $22.50.

3. A group of people wish to hire a bus to go to a festival. The cost per person ($C) is inversely reated to the number of people on the bus (N). So the more people who travel on the bus, the cheaper the cost per person.

If 25 people travel, the cost per person is $24.

(i) Develop an equation to link cost per person to the number of people travelling.

(ii) How much does it cost per person if 8 people travel?

(iiI) How many people would travel if the cost per person reduced to $10.

Answer.(i) C = 600/n.
(ii) Cost = $75 per person
(iii) 60 people.

4. The time (T) it takes for water to flow into a container varies inversely with the rate at which water is flowing (R).

It takes 42 minutes to fill a tank when water flows at 25 litres per hour.

(i) Develop an equation to link the time taken to the amount of water (W) in a container.

(ii) How long will it take the tank to fill if the rate of flow is increased to 35 litres per hour?

(iii) Use your equation to determine the rate of flow if the tank is to be filled in one hour.

Answer.(i) W = 25T.
(ii) T = 30 mins.
R = 12.5 l/hour.

5. The time (in hours) it takes for an accounting department to balance the books varies inversely with the number of accountants working on the task. Last year, when 4 accountants worked on balancing the books, the task took 20 hours to complete.

(i) Show that the relationship between h, the number of hours to complete the task, and n, the number of accountants working on the task, can be given by the equation .

(ii) How many hours would it take if twice as many accountants work on the task this year than last year.

(iii) How many accountants would need to be on the task for it to be completed in 8 hours?

(iv) Sketch a graph of the relationship between h and n.

Answer.(i)k = 80.
(ii) 10 hours.
(iii) 10 accountants.

6. The number of seats in a row at a cinema varies inversely with the width of each seat.

The present arrangement has 42 seats in each row. Each seat is 47 cm wide and this arrangement uses up all of the allowed space to comply with safety regulations.

The owners of the cinema are planning to remodel the cinema.

(i) How many seats of width 52 cm can be installed?

(ii) If only 34 seats are required in each row, what is the maximum width of each seat?

Answer.(i)37 seats.
(ii) 58 cm wide.

7. A car travels at a constant speed. The time taken for a journey between two locations varies inversely with its speed.

It takes 2.5 hours for the car to travel between the two towns at a constant speed of 60 km/h.

(i) Calculate the distance between the two towns.

(ii) Develop an equation linking time and speed for the journey.

(iii) How long does the journey take if the car's speed is 50 kph.

(iv) By plotting four points, draw the curve that shows the time taken to travel between the two towns at different constant speeds.

Answer.(i) Distance = 150 km.
(ii) T = 150/S.
(iii) 3 hours.

8. The time (t) required to empty a tank varies inversely with the rate of pumping (r).

A pump can empty a tank in 8 hours if its rate of pumping is 250 litres per hour.

(i) How long will the pump take to empty the same tank if the rate is reduced to 150 litres per minute?

(ii) What rate would be required to empty the tank in 5 hours?

Answer. k = 2000:
(i)13 hours 20 mins.
(ii) Rate is 400 litres/hour.

9. The pressure (P) required to break a plank of timber varies inversely with the length (L) of the board.

A 2 metre long board requires the equivalent of 120 kgs of pressure.

(i) What pressure is required to break a 1.5 m board?

(ii) What length of board could be broken if the pressure was the equivalent of 200 kgs?

Answer.K = 240
(i)160 kgs.
(ii) 1.2 m board.

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