This is for grade 11 math, under quadratic extrema and roots.
Fred and Ginger have 24 m of fencing to enclose a garden at the back of their house. From the pictures you can see (shows a large rectangle for the house, with a smaller one attached on the bottom line, to the right) they want to fence only three sides of the garden. What are the dimensions of the garden having the biggest area they can enclose with their fence? What is the maximum area?
Hint: Define x to be the garden's width and get an equation for the part of the perimeter being fenced. Put this expression in an area formula and notice that your area formula becomes a quadratic function in x. (You know how to find the maximum of a quadratic function.)
Here's what I got so far.
Let x be the width.
Let L be the length.
Let A be the area.
2x + L = 24
L = 24 - 2x
A = Lx
A = (24 - 2x)X
= -2x(squared) + 24x
This is with my teacher's help. I just need to know how to get the answers, what my next steps should be, etc.
well... break it down, first focus on 2x + L = 24
L = 24 - 2x
this should be your first step.
What I did with this problem is that I rearanged the problem so it was easy.
First ask youself what would be the max number that I could use to make x= a number times 2 = another number to = 24. I don't think this is the max you can go but so far i got L= 24- 2(2)
Great job on setting up the equations! Now, to find the dimensions of the garden that will maximize the area, we need to find the value of x that corresponds to the maximum point of the quadratic function A = -2x^2 + 24x.
To find this maximum point, you can follow these steps:
Step 1: Start with the quadratic function: A = -2x^2 + 24x
Step 2: Notice that the coefficient of x^2 term is negative (-2). This indicates that the graph of the quadratic function will be a downward-opening parabola.
Step 3: To find the x-coordinate of the vertex (maximum point) of the parabola, you can use the formula x = -b / (2a). In this case, a is the coefficient of x^2, which is -2, and b is the coefficient of x, which is 24.
x = -24 / (2 * -2)
x = -24 / -4
x = 6
Step 4: Now that you have the x-coordinate of the vertex, substitute this value back into the equation for A to find the maximum area. So, let's substitute x = 6 into the equation A = -2x^2 + 24x:
A = -2(6)^2 + 24(6)
A = -2(36) + 144
A = -72 + 144
A = 72
Therefore, when the width (x) of the garden is 6 meters, the area (A) of the garden is maximized, and its maximum area is 72 square meters.
So, the dimensions of the garden that have the biggest area they can enclose with their fence are a width of 6 meters and a length of 24 - 2(6) = 12 meters.