Optimisation

• Find the roots, the answer will be the root that is not at (0,0).

• Find the vertex by completing the square

• Find the vertex via the midpoint of the roots

• Find the y-intercept, multiply it by -5, and add 100.

A rocket's height in meters is modelled by h(t) = -5t^2 + 100t, where t is measured in seconds.

Find the value of the maximum height, and the time at which it attains this height.

• t = 10s, h = 500m

• t = 20s, h = 250m

• t = 5s, h = 375m

• t = 15s, h=375m

We are to create a yard using 180 meters of bushes, that borders a river on one side (see image).

Use the variables given in the image to write equations for the perimeter (P) and area (A) of this yard.

• A=xy

  P=2x+2y=180

• A=xy

  P=2x+y=180

• A=x^2y

  P=2x+2y=180

• A=x^2y

  P=2x+y=180

Select the most suitable option to express the area (A) in terms of one variable x. Remember that a total of 180m of fence is allowed.

• A = 2x^2-180x

• A = x(A/x)

• A = x(180-2x)

• A = 180x-x^2

The equation was A = x(180-2x). Suggest an appropriate x-coordinate for the vertex. (Hint: the roots are x=0 and something else...)

• Vertex at x=60

• Vertex at x=30

• Vertex at x=90

• Vertex at x=45

A = x(180-2x)

The x-coordinate of the vertex is 45.

Use that information, and find the maximum area of the yard.

• 180m^2

• 4050m^2

• 2025m^2

• 6075m^2

Now instead of building a yard that borders a river, you want to build two IDENTICAL yards bordering each other.

What do you think the first step is, if you want to find out how to maximise the area of both yards combined?

(You will have 120m of fence)

• Write an equation for perimeter and area

• Sketch a graph of y vs x

• Find the vertex

• A = x(120-x)

Considering the diagram attached, express the TOTAL area of BOTH yards (A), and the total fencing used (P) in terms of x and y.

Remember you have 120m of fence.

• P = 4x+3y=120

  A = x^2y

• P = 4x+4y=120

  A = 2xy

• P = 4x+3y=120

  A = 2xy

• P = 2x+3y=120

  A = x^2y

I am thinking of two numbers adding up to 2. I would like to find out their maximum product.

What is the most reasonable first step to do?

• Find the vertex

• Write an equation for area and perimeter in terms of x and y, then find the vertex

• Solve the equation for x+y=2 by using substitution

• Write an equation relating the sums and products of two numbers using variables

I am thinking of two numbers adding up to 2. I would like to find out their maximum product.

Choose the most suitable equations describing this scenario.

• x+y=2

  Product = xy

• x+y=2

  Product = x^2

• x+y=2

  xy = 1

• x+y=2

  x(2-x) = 1