Question

A rectangle has been drawn in such way that one vertex is on the beginning of coordinates plane, two sides are on the y and x axis, and another vertex is a point in f(x) = 4 - 2x/3. What are rectangle vertexes coordinates so that the area of the rectangle is maximum?

Answer 1

My attempt to solve this:
The question is basically asking what is the maximum number that can be obtained when x*y (x and y represent the point in the function that would be a rectagle's vertex).
Function touches y axis in point (0, 4) and x axis in point (6, 0)
I think we need to find the middle point between x and y? So, it would be (3, 2)

Am I right and how is it the case? (My reasoning is that the area would go lower if x or y got bigger, but I got this fact from trying different values, so it's not really solid...)

Answer 2

Nice to see you didn't jump to calculus the moment you saw the word "maximum"

Answer 3

That might be because I have very little knowledge in calculus yet ;)

Answer 4

But I wonder how to prove that my solution is right / wrong...

Answer 5

Area = x*y = x*(4-2x/3)

Answer 6

You must be knowing how to find the vertex point of a parabola w/o using calculus.

Answer 7

I am not sure what you mean, but I think that it is possible to draw a parabola g(x)=4x - 2x^2/3 and the intersection point of f(x)= 4 - 2x/3 would be the vertex point? So, basically a system of equations and then a quadratic?

Answer 8

No. Its very simple. You're just not looking at it correctly.

Area expression is given by x(4-2x/3)

Answer 9

Fine with that expression ?

Answer 10

Yes

Answer 11

What geometric shape does the graph of the function A(x) = x*(4-2x/3) represent ?

Answer 12

parabola

Answer 13

What kind of parabola, facing up or facing down ?

Answer 14

facing down

Answer 15

So it will have a MAXIMUM value at the vertex.
Can you find the vertex ?

Answer 16

Well, vertex would be x = 0, so y = 0?

Answer 17

No. Vertex is the turning point :

Answer 18

Haven't you studied parabolas before ?

Answer 19

I have, but this parabola has symmetry from y axis, so the vertex would be in (0, c). So, x = 0..

Answer 20

Oh, yes. I got you. You're right. Find y coordinate also

Answer 21

If x = 0, then y = 0 as well. (0,0)

Answer 22

Hey no. Sorry I wasn't paying attention. You're wrong. (0,0) is not the vertex

Answer 23

But how comes? :D
A(x)= x(4 - 2x/3)
So, if x = 0, then whole RHS would be 0 and that would make y = 0

Answer 24

That means (0, 0) is a point on the parabola. It need not be the vertex :