DP IB Maths: AA SL

Topic Questions

5.5 Optimisation

1a
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2 marks

A company manufactures food tins in the shape of cylinders which must have a constant volume of 150 pi cm cubed.  To lessen material costs the company would like to minimise the surface area of the tins.

By first expressing the height h of the tin in terms of its radius r, show that the surface area of the cylinder is given by S equals 2 pi r squared plus fraction numerator 300 pi over denominator r end fraction.

1b
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4 marks

Use calculus to find the minimum value for the surface area of the tins. Give your answer correct to 2 decimal places.

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2a
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2 marks

A right-angled triangle of height h, base r and hypotenuse 15 space cm has been rotated about its vertical axis to form a cone.

q2a-5-5-medium-ib-aa-sl-maths

Write an expression for r in terms of h.

2b
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3 marks

Show that the volume of the cone, V space cm cubed, can be expressed as:

V equals straight pi over 3 left parenthesis 225 h minus h cubed right parenthesis

2c
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3 marks

Find the value of h which provides the cone with its maximum volume.

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3a
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3 marks

A wire of length 1 m is cut into two pieces. The first piece is bent into the shape of a square. The second piece is bent into a rectangle which has a length l twice as long as its width w. Let x cm be the perimeter of the square.

Find an expression for the area of the square.

3b
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3 marks

Show that the width of the rectangle w equals fraction numerator 100 minus x over denominator 6 end fraction.

3c
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3 marks

Find an expression for the sum of the area of the two shapes, S.

3d
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4 marks

Find the value of x such that the sum of the areas, S, is a minimum.

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4a
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2 marks

Liam, a keen runner and swimmer, enters a competition which requires the competitors to run from point A along a straight beach, before diving into the water and swimming directly to point C. Liam is able to run at a speed of 8 m/s along the beach and swim at 2 m/s in the water.

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Let x represent the distance between A and B,  the distance that Liam runs along the beach before entering the water and swimming along the line BC.

Find an expression for the time taken for Liam to run x metres between A and B.

4b
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2 marks

Show that the length of B C equals square root of 10000 plus left parenthesis 500 minus x right parenthesis squared end root.

4c
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2 marks

Find an expression for the total time taken for Liam to finish the race.

4d
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3 marks

Find the value of x that will allow Liam to complete the race in the quickest time.

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5a
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2 marks

A small cylindrical drum, closed at the top but open at the bottom, has a radius r cm and a height h spacecm. The volume of the drum is 1000 space cm cubed.

The material to make the top skin of the drum costs 25 cents per cm squared and the curved surface of the drum costs 20 cents per cm squared.

Find an expression for h in terms of r.

5b
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4 marks

Show that the total cost of the material to make the drum is C equals 25 pi r squared plus 40000 over r.

5c
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2 marks

Find fraction numerator d C over denominator d r end fraction.

5d
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4 marks

The function C open parentheses r close parentheses has a local minimum at the point left parenthesis p comma space q right parenthesis.

Find the value of p.

5e
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1 mark

State, to the nearest dollar, the minimum cost required to make the drum.

5f
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4 marks

Find begin inline style fraction numerator d squared C over denominator d r squared end fraction end style and hence, describe the concavity of the function C open parentheses r close parentheses at x equals p.

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6a
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1 mark

The daily cost function of a company producing pairs of running shoes is modelled by the cubic function

C left parenthesis x right parenthesis equals 1225 plus 11 x minus 0.009 x squared minus 0.0001 x cubed comma space space space space space space space space 0 less or equal than x less than 160

where x is the number of pairs of running shoes produced and C the cost in USD.

Write down the daily cost to the company if no pairs of running shoes are produced.

6b
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2 marks

The marginal cost of production is the cost of producing one additional unit. This can be approximated by the gradient of the cost function.

Find an expression for the marginal cost, C to the power of apostrophe left parenthesis x right parenthesis, of producing pairs of running shoes.

6c
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2 marks

Find the marginal cost of producing

(i)
40 spacepairs of running shoes
(ii)
90 spacepairs of running shoes.
6d
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3 marks

The optimum level of production is when marginal revenue, R to the power of apostrophe left parenthesis x right parenthesis, equals marginal cost, C to the power of apostrophe left parenthesis x right parenthesis. The marginal revenue, R to the power of apostrophe left parenthesis x right parenthesis, is equal to 4.5.

Find the optimum level of production.

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7a
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2 marks

A cyclist riding over a hill can be modelled by the function

h left parenthesis t right parenthesis equals negative 1 over 24 t squared plus 3 t plus 12 comma space space space space space space space space space space 0 less or equal than t less or equal than 70

where h is the cyclist’s altitude above mean sea level, in metres, and t is the elapsed time, in seconds.

Calculate the cyclist’s altitude after a minute.

7b
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2 marks

Find h to the power of apostrophe left parenthesis t right parenthesis.

7c
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3 marks

Calculate the cyclist’s maximum altitude and the time it takes to reach this altitude.

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8a
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1 mark

A company produces and sells cricket bats. The company’s daily cost, C, in hundreds of Australian dollars (AUD), changes based on the number of cricket bats they produce per day. The daily cost function of the company can be modelled by

C left parenthesis x right parenthesis equals 6 x cubed minus 10 x squared plus 10 x plus 4

where x hundred cricket bats is the number of cricket bats produced on a particular day.

Find the cost to the company for any day zero cricket bats are produced.

8b
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2 marks

The company’s daily revenue, of AUD, from selling x hundred cricket bats is given by the function R left parenthesis x right parenthesis equals 42 x.

Given that profit = revenue - cost, determine a function for the profit, P left parenthesis x right parenthesis comma in hundreds of AUD from selling x hundred cricket bats.

8c
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2 marks

Find P to the power of apostrophe left parenthesis x right parenthesis.

8d
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3 marks

The derivative of P open parentheses x close parentheses gives the marginal profit. The production of bats will reach its profit maximising level when the marginal profit equals zero and P open parentheses x close parentheses is positive.

Find the profit maximising production level and the expected profit.

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9a
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2 marks

Dora decides to build a cardboard container for when she goes strawberry picking from a rectangular piece of cardboard, 55 space cm space cross times 28 space cm. She cuts squares with side length x cm from each corner as shown in the diagram below.

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Show that the volume, V space cm cubed, of the container is given by

V equals 4 x cubed minus 166 x squared plus 1540 x

9b
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2 marks

Find begin inline style fraction numerator d V over denominator d x end fraction end style.

9c
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4 marks

Find

(i)
the value x of that maximises the volume of the container
(ii)
the maximum volume of the container. Give your answer in the form a cross times 10 to the power of k, where 1 less or equal than a less or equal than 10 and k element of straight integer numbers.

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