The sum of a number and two times another number is 24. Find the numbers if their product is a maxiumum and state the maximum.
Please explain it because i don't know how to do these kind of questions... I have a test tomorrow...plz help..
The sum of a number and two times another number is 24. Find the numbers if their product is a maxiumum and state the maximum.
Please explain it because i don't know how to do these kind of questions... I have a test tomorrow...plz help..
Start with the basics. what are our unknowns here? the two numbers of course. so, they're unknowns, call them something.
Let one number be $\displaystyle x$
Let the other number be $\displaystyle y$
ok, what are we told about x and y?
the sum of $\displaystyle x$ and twice $\displaystyle y$ is 24. "sum" means "add". so we have:
$\displaystyle x + 2y = 24$ .......................(1)
ok, what now? We want their product to be a maximum. call the product $\displaystyle P$. product means to multiply, so we want:
$\displaystyle P = xy$ .............(2) to be a maximum
so we have our constraint, which is equation (1), and we have the function we want to maximize, which is $\displaystyle P$
now, we can take two routes to complete the question. which route we take depends on whether you can use calculus or not. so, can you?
ok, so we want the precalc way of doing this. that's fine.
so we have $\displaystyle x + 2y = 24$ ..........(1)
and $\displaystyle P = xy$ ................(2)
we want to maximize P. to do this, it would be much simpler to work with one variable. so solve for one variable in terms of the other using our constraint.
From equation (1), $\displaystyle x = 24 - 2y$, so plug that into P
$\displaystyle \Rightarrow P = (24 - 2y)y = 24y - 2y^2$
this is a downward opening parabola, it's maximum value occurs at the vertex. we can find the vertex by completing the square, but that is too much work. use the vertex formula:
Vertex formula: for a parabola $\displaystyle y = ax^2 + bx + c$, the x value for the vertex is given by: $\displaystyle x = \frac {-b}{2a}$, and thus the vertex is given by $\displaystyle \left( \frac {-b}{2a}, f\left( \frac {-b}{2a} \right) \right)$
so here, the vertex occurs when $\displaystyle y = \frac {-24}{2(-2)} = 6$
so $\displaystyle \boxed {y = 6}$
but we have $\displaystyle x = 24 - 2y$
$\displaystyle \Rightarrow x = 24 - 2(6) = 12$
$\displaystyle \Rightarrow \boxed {x = 12}$
there are your two numbers
now, can you state the maximum product?
thnx... the max value is 72... i'm really new to this method (srry i don't get it) ... i used the equations u made... after that i completed the square... and got it in the form y = a(x-h)^2 + K
so my answer was
= -2 (y^2 -6)^2 + 72
and 72 was the max... my teacher told me that 72 is the max when x-coordinate is -6... so the other number will be "-12"
so is my answer right?