maxima and minima probs... (1 Viewer)

[keropi]

hey
could anyone please show me how to do the folowing questions...

1. Find two numbers whose sum is 28 and whose products is a maximum.

2. Bill wants to put a small rectangular vegetable garden in his backyard using two existing walls as part of its border. He has 8m of garden edging for the border on the other two sides. Find the dimensions of the garden bed that will give the freatest area?

thanks....

 

[insert-username]

First off, hey there, and welcome!

1. Find two numbers whose sum is 28 and whose products is a maximum.

From the question, you know that:

x + y = 28 (sum is 28)

Thus y = 28 - x

xy = M (some maximum value)

Now you need to substitute in your value for y in terms of x (28 - x). To then work out the maximum value, derive the equation, and since it's a concave down parabola (with a negative x²), you can work out its maximum point using the derivative - there will be a stationary point at the maximum value.

x(28-x) = M

28x - x² = M

dM/dx = 28 - 2x

Therefore stationary point at x = 14

When x is 14, y is 14

So the values are 14 and 14.

2. Bill wants to put a small rectangular vegetable garden in his backyard using two existing walls as part of its border. He has 8m of garden edging for the border on the other two sides. Find the dimensions of the garden bed that will give the freatest area?thanks....

Since he has 8m of garden edging, the two sides that aren't part of the wall must be equal to 8m. Let x and y be the two sides:

x + y = 8

xy = M

Now follow exactly the same procedure as I did above. Write y in terms of x, substitute, derive, and work out stationary point and hence maximum value.

Hope that helps,


I_F

 

[keropi]

thanks for ur help