Let a and b be two positive numbers. if 2a + 3b=6 then the maximum product of these a and b is: a: 1 b: 1.5 c: 3 d: 4

a= 3/2

b= 1

a*b= 3/2

Step-by-step explanation:

For getting this problem we need to think in terms of only one unknown. Lets start by looking at our equation:

2a + 3b=6

No, we need to see what happens to a*b. Lets get the value of one of these unknowns in terms of the other, this is, a in terms of b or b in terms of a. Here, I will get a in terms of b (if you like, do the other way and compare):

2a + 3b=6

Subtract 3b in both sides:

2a = 6 - 3b

Now divide both sides by 2 to get a alone:

a = (6 - 3b)/2

a = 3 - 3/2 b

Now lets try to multiply a by b, using this a we found above:

a*b = (3 - 3/2 b)*b

Using distributive:

a*b = 3b - 3/2 b^2

So, we have an expression for a*b that depends only on b. Notice that this expression is a parabola with negative coefficient on the square term, what makes it a negative parabola that MUST have a maximum value. So, we have an expression for a*b that can be maximized, so we can find the maximum of a*b by the derivative of the expression. Lets derive in b:

(3b - 3/2 b^2)' = (3b)' - (3/2 b^2)' = 3 - 3b

So, the derivative equal to 0 gives us the maximum:

3 - 3b = 0

Suming 3b in both sides:

3 = 3b

Dividing by 3:

1 = b

So, we maximize our expression a*b when b is equal to 1. Now, we can replace it on a = 3 - 3/2 b to find b:

a = 3 - 3/2(1) = 3 - 3/2 = 3/2

Thus, a is equal to 3/2 and the product a*b is maximized when:

a = 3/2

b = 1

And the product is a*b= (3/2) * 1 = 3/2

hey there!

if she's saving for six months, let's divide the total number we need for the trip by six, dividing it into the amount that needs to be saved for each month.

500 ÷ 6 =

she needs to save at least this amount each month, so we should use this symbol: ≥

let x be the amount she saves each month.

your final answer is x ≥

hope this !