Maximum / Minima Problems (1 Viewer)

[Jmmalic220]

6. Find two numbers whose sum is 28 and whose product is a maximum.

8. A piece of wire 10m long is broken into two parts, which are bent into the shape of a rectangle and a square. Find the dimensions x and y that make the total area a maximum. (x = width of rectangle, y = length of rectangle; x is also = side of square)

9. A box is made from an 80 cm by 30 cm rectangle of cardboard by cutting out 4 equal squares of side x cm from each corner. The edges are turned up to make an open box. The volume of this box is given by the equation V=4x³-220x²+2400x.

(a) Find the value of x that gives the box its greatest volume.

11. A silo in the shape of a cylinder is required to hold 8600m³ of wheat.
(a) Find an equation for the surface area of the silo in terms of the base radius.
(b) Find the minimum surface area required to hold this amount of wheat, to the nearest square metre.

Please show working. Thank you very much.

 

[Squar3root]

Ill use the 1st question as an example and you try the rest.

1. Find a constraining equation/s

in this case let one of the numbers be x. Since their sum is 28 the other number must be (28-x). Now you want the product to be maximum of these 2 numbers which would be y =x (28-x) where y is the maximum. Expansion of this is y = 28x -x^2

Now to find the maximum, you need to take the derivative and set it equal to zero. From there you need to test that it is a maxium by using the table method or substitution into the second derivative. in this case x = 14

have a go at the others by yourself

 

[davidgoes4wce]

Squareroot did the question right but I had a go at doing it my way:

 

[davidgoes4wce]

Q8.

 

[Squar3root]

Squareroot did the question right but I had a go at doing it my way:

 

[davidgoes4wce]

Defining P as the total Area of the shape.

Happy?

 

[davidgoes4wce]

A lecturer by the name of Alec Lee told me we should always define out variables before we do differentiation and integration. Good lecturer that man.

 

[Squar3root]

Defining P as the total Area of the two shapes.

Happy?

iirc when i was doing 2U, out math teacher said he would not give the mark if you just "magically" equated the derivative to zero. He said you're meant to write why that is the case "to locate maximum or minimum points we must find dP/dx = 0"

 

[davidgoes4wce]

iirc when i was doing 2U, out math teacher said he would not give the mark if you just "magically" equated the derivative to zero. He said you're meant to write why that is the case "to locate maximum or minimum points we must find dP/dx = 0"

Fair point.

Alec Lee who is a university lecturer also told me we should define our variables. I remember he did it for every calculus application question, inverse trig function etc.................

 

[davidgoes4wce]

Do you happen to know Alec Lee? Im not sure how well he is known in HSC circles

 

[davidgoes4wce]

Q9.

V= Volume
L=Length
W=width
H=Height
x= represents the corner lengths

 

[Squar3root]

I haven't heard of him

 

[davidgoes4wce]

Anyway I've taken that point on board.

I think calculus your always learning new things along the way. In this question I've put that statement in before setting the derivative equal to 0.

 

[He-Mann]

I think calculus your always learning new things along the way.

Not just calculus. Life in general.