Maximum and minimum problems Calculus AS 3.6

A rectangular block is constructed so that its length is twice its breadth. Find the least possible surface area of the block if its volume is 72

Step 1: Draw a diagram and label it “its length is twice its breadth” x 2x2x h

Step 2: Write the equation for the relationship given its volume is 72 x 2x2x h

Step 3: Write the equation for what you are trying to maximise or minimise “Find the least possible surface area” x 2x2x h

Step 4: Use the information to write this equation in one variable only “Find the least possible surface area”

Step 5: Differentiate = 0 “Find the least possible surface area”

Step 6: Answer the question “Find the least possible surface area”

IF the size of a parcel sent through the post is limited by the fact that the sum of its length and girth must not exceed 2 m, find the volume of the largest rectangular parcel with a square base which may be posted.

1. Draw a diagram x x y

2. Write the equation for the relationship given x x y

3. Write the equation for what you are trying to maximise or minimise x x y find the volume of the largest rectangular parcel

4: Use the information to write this equation in one variable only x x y find the volume of the largest rectangular parcel

5: Differentiate = 0 find the volume of the largest rectangular parcel

6: Answer the question find the volume of the largest rectangular parcel Don’t forget units

A feeding trough is made from three pieces of metal welded together, one rectangle of size 3k cm by 5k cm and two trapeziums. The length of the trough is 5k cm and its cross- section is shown in the diagram. (k is constant)

kk k Find the maximum volume of the trough. (5k long)

kk k Diagram is given and we only need to use 1 variable, θ (5k long)

kk k

kk k k

kk k k

kk k k

kk k k

kk k k

kk k k

kk k Find the maximum volume of the trough. k

A circle of radius l is used to form a cone. Find the angle of the sector that forms a cone of the maximum volume.

Think about how the cone is created θ l l l

The arc length is the same as the circumference of the rim of the cone. θ l l l r

Using Pythagoras’ theorem θ l l l r h

Volume of the cone = θ l l l r h

Substituting

Clean it up

Differentiate using product rule

Would give a minimum

Differentiate using product rule

If total revenue for a firm is given by where x is the number of units sold, find the number of units that must be sold to maximize revenue.

The number of units that must be sold to maximize revenue is 40.

A travel agency will plan a group tour for groups of 25 or larger. If the group contains exactly 25 people, the charge is $300 per person. However, each person’s cost is reduced by $10 for each additional person above the 25. What size group will produce the largest revenue for the agency?

A travel agency will plan a group tour for groups of 25 or larger. If the group contains exactly 25 people, the charge is $300 per person. However, each person’s cost is reduced by $10 for each additional person above the 25. What size group will produce the largest revenue for the agency? R = number of people x cost

Group size 22 or 23 for revenue $7560

Suppose the production capacity for a certain commodity cannot exceed 30. If the total profit function for this commodity is where x is the number of units sold, find the number of items that will maximize profit.

Suppose the production capacity for a certain commodity cannot exceed 30. If the total profit function for this commodity is where x is the number of units sold, find the number of items that will maximize profit. Check end point x = 30, P = $26,800

End point x = 30, P = $26,800 gives the maximum

A man at a point A on the shore of a circular lake with radius 2 km wants to be at point C diametrically opposite A on the other side of the lake in the shortest possible time. The ratio of his walking speed to his rowing speed is 2:1. At what angle to the diameter should he row?

Check end points

Time to row across A C 4 km

Time to walk around A C 4 km

Time to row and then walk: A C 4 km DRDR θ DwDw

Angle in a semi-circle means we have a rt angled triangle A C 4 km DRDR θ

Length of the arc A C 2 km DwDw θ 2θ

Differentiate

Differentiate and solve

Walking around the lake takes the least time

My boat travels at 30 km/hr and, because I never speed, I travel at 50 km/hr by car along the coast. At what point should I keep the car to minimize the time travelling to town. 8 km town 48 km

Evaluate the end points: My boat travels at 30 km/hr 8 km town 48 km

Evaluate the end points: Boat then car 8 km town 48 km

Evaluate: Boat to x, then car 8 km town (48 – x) km x

Evaluate: Boat to x, then car 8 km town (48 – x) km x

Evaluate: Boat to x, then car

8 km town (48 – x) km x

The cost of fuel per hour of running a jet aircraft, $C, is a function of its cruising velocity v km/hr. Find the most economical cruising velocity for a journey of 6000 km.

The cost of fuel per hour of running a jet aircraft, $C, is a function of its cruising velocity v km/hr. Find the most economical cruising velocity for a journey of 6000 km.

The cost of fuel per hour of running a jet aircraft, $C, is a function of its cruising velocity v km/hr. Find the most economical cruising velocity for a journey of 6000 km.

The cost of fuel per hour of running a jet aircraft, $C, is a function of its cruising velocity v km/hr. Find the most economical cruising velocity for a journey of 6000 km.

The cost of fuel per hour of running a jet aircraft, $C, is a function of its cruising velocity v km/hr. Find the most economical cruising velocity for a journey of 6000 km.

The total cost per hour of running a ship while on a voyage is Where v is the constant speed for the voyage in km per hour. If the ship makes a voyage of 2000 km, find the speed which gives the most economical cost for the voyage.

A trapezium is inscribed in a semicircle of radius r so that one side of the trapezium is on the diameter of the semicircle. Find the maximum area of the trapezium, in terms of r.

Find the maximum area of the trapezium, in terms of r.

A rectangle is drawn as shown. The equation of the parabola is Find the greatest area that this rectangle can have.

A rectangle is drawn as shown. The equation of the parabola is Find the greatest area that this rectangle can have.

A rectangle is drawn as shown. The equation of the parabola is Find the greatest area that this rectangle can have.

A cone has a height of 15 cm and radius of base of 9 cm. Another smaller cone is inscribed in the cone. Its vertex is at the centre of the base of the larger cone. The base of the smaller cone is parallel to the base of the larger cone. Find h and r, the dimensions of the smaller cone, so that it has maximum volume. State the maximum volume of the smaller cone.

Find h and r, the dimensions of the smaller cone, so that it has maximum volume. State the maximum volume of the smaller cone.