Question

 

1-The sum of two numbers is 34.
    a)Find the largest possible product of these numbers.
    b)What would be the largest possible product if the sum if the two numbers were "k"?

2-Sixty meters of fencing are used to fence a rectangular garden.
    a)Find the dimensions that will give that maximum area.
    b)What would be the maximum area if "k" feet of fencing were used in terms of "k"?

THANK YOU

Answer 1

Solution:

(1). Let the first number be x. Since, sum of the two number is 34. So, second number would be 34-x.

(a). The product (P) of these two numbers is

For maximum P, dP/dx=0. So,

So, the first number is 17 and second number is 34-17=17

Larget product is

(b). Let first number be x, then second number is k-x. The product (P) is.

For maximum P,

  

So, the numbers are k/2 and (k-k/2)=k/2.

The maximum product,

(2).  Let the length of the rectangle=x meters and width=y meters.

Since, perimeter (fencing) of the garden is 60 meters. So,

2(x+y)=60

x+y=60/2

x+y=30

y=30-x meters.

(a).  The area (A) of the rectangle is

For maximum, P

So, width=30-15=15 meters.

Hence, for maximum area,the length=15 meters and width=15 meters.

(b). If the perimeter is 'k'. Then,

The area (A) is

For maximum area,

So, length=k/2 and width=(k/2-k/4)=k/4.

So, the maximum are is