11I.2 - Optimisation

ABC

11I.2 - VIDEO EXAMPLE 1:

An open box is to be constructed from a rectangular piece of cardboard measuring 100 cm × 120 cmby cutting out four equal square from each of the corners.

• Draw a diagram to represent the construction of the box.
• Find and expression for the volume of the box in terms of x.
• State the practical domain for x.
• Find the largest possible volume the box can have. State your answer correct to 2 decimal places.

11I.2 - VIDEO EXAMPLE 2:

A popular type of window is the Norman Window which has a semi-circle on top of a rectangle, as shown to the right. The total perimeter of the window is 8 metres.

• Find an expression for ℎ in terms of 𝑟.
• Hence, find an expression for the area of the window, 𝐴, in terms of 𝑟.
• Find the maximum possible area that the window can have.
• State your answer correct to 2 decimal places.

11I.2 - VIDEO EXAMPLE 3:

Louie Gee launches a boat from point 𝐴 on the bank of a uniform, straight river that is 4 km wide. He wants to reach point 𝐵, which is located 10 km downstream on the opposite side of the river, as quickly as possible. He could row his boat directly across the river to point 𝐶 and then run to 𝐵, or he could row directly to 𝐵, or he could row to some point 𝐷, between 𝐶 and 𝐵, and then run to 𝐵. Louie Gee is able to row at a speed of 6 km/h and run at a speed of 8 km/h.

• Give an expression for the time, 𝑇, that it takes Louie Gee to travel from point 𝐴 to point 𝐵 in terms of 𝑥.
• Find the position, 𝑥, where Louise Gee should land to minimise the time taken to travel from point 𝐴 to the point 𝐵.
• Hence, state the minimum amount of time, in hours, that it takes for Louie Gee to complete his journey. State your answer correct to 2 decimal places.

11I.2 - VIDEO EXAMPLE 4:

A soup company needs to make a cylindrical tin can that has a capacity of 500 𝑚𝐿 of soup. Find the dimensions of the can that will minimise the costs of production.

• Show that the surface area, 𝐴 𝑐𝑚², of the tin is: