it takes 80 feet of fencing to enclose your garden. According to your calculations,you will need 350 square feet to plant everything you want. Is your garden big enough? Explain your answer.
They gave a rectangle length=40-x and width= x
I asked my mom and she also got confused.So we are stuck.
Well, don't worry, I've got your back! Let's figure this out together, and remember, with me around, we'll have a fun time doing it!
Alright, to determine if your garden is big enough, we need to consider both the perimeter (fencing) and the area (planting space).
First, let's talk about the perimeter. You mentioned that it takes 80 feet of fencing to enclose your garden. Since the garden is in the shape of a rectangle with a length of 40-x and a width of x, we can calculate the perimeter using these dimensions.
The perimeter of a rectangle is given by the formula P = 2(l + w), where P is the perimeter, l is the length, and w is the width.
So in this case, the perimeter of your garden would be P = 2((40-x) + x) = 2(40) = 80 feet. Bingo! The perimeter matches, which means we're on the right track.
Now let's move on to the area. You mentioned that you need 350 square feet of planting space. To find the area of a rectangle, we use the formula A = l * w, where A is the area, l is the length, and w is the width.
So, A = (40-x) * x = 40x - x^2. Great!
Now, if your desired planting space is 350 square feet, we can set up an equation: 40x - x^2 = 350. Don't panic, solving equations can be tricky, but we'll get through it together!
Unfortunately, I can't solve this equation for you without knowing the exact values of x, but you can use algebraic methods like factoring, completing the square, or the quadratic formula to find the values of x that satisfy the equation. Once you find those values, you can see if they make sense in the context of your garden dimensions.
So, in a nutshell, the size of your garden depends on whether the values of x that satisfy the equation 40x - x^2 = 350 allow for reasonable and feasible dimensions for your garden. Talk to your mom and give it another shot. I hope this helps, and remember, a little humor can go a long way when trying to solve a problem!
To determine if your garden is big enough, we need to compare the area of your garden to the required planting area.
The area of a rectangle is given by the formula: Area = length * width.
In this case, the length of your garden is defined as 40 - x and the width is x.
Therefore, the area of your garden is: Area = (40 - x) * x.
Since you mentioned that you need 350 square feet to plant everything you want, we can set up the following equation:
(40 - x) * x = 350.
Now, let's solve this equation step by step:
Expand the equation:
40x - x^2 - 350 = 0.
Rearrange the equation:
-x^2 + 40x - 350 = 0.
To make the equation easier to work with, let's multiply both sides by -1:
x^2 - 40x + 350 = 0.
Now, we can solve this quadratic equation. If we factorize it, we get:
(x - 10)(x - 35) = 0.
This equation will be true if either (x - 10) = 0 or (x - 35) = 0.
Solving these equations, we find that x = 10 or x = 35.
So your options for the width of your garden are either 10 feet or 35 feet.
Now, to determine if your garden is big enough, we need to check the area for each option:
For x = 10:
Area = (40 - 10) * 10 = 30 * 10 = 300 square feet.
For x = 35:
Area = (40 - 35) * 35 = 5 * 35 = 175 square feet.
Since you mentioned that you need 350 square feet to plant everything, neither of these options is sufficient. This means your garden is not big enough to accommodate everything you want to plant.
You may need to consider a larger area or find a way to optimize your planting layout to fit within the available space.
To determine if your garden is big enough, you need to compare the area of the garden to the desired planting area.
The formula for the area of a rectangle is: Area = length * width
In this case, the length of the rectangle is given as 40 - x, and the width is x.
So, the area of the rectangle can be calculated as: Area = (40 - x) * x
Given that the desired planting area is 350 square feet, we can set up an equation to solve for x:
(40 - x) * x = 350
Simplifying the equation, we get:
40x - x^2 = 350
Rearranging, we have:
x^2 - 40x + 350 = 0
To solve this quadratic equation, we can use factoring, completing the square, or the quadratic formula. In this case, factoring is not straightforward, so let's use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
For our equation, a = 1, b = -40, and c = 350.
Plugging these values into the quadratic formula, we get:
x = (-(-40) ± √((-40)^2 - 4(1)(350))) / (2(1))
x = (40 ± √(1600 - 1400)) / 2
x = (40 ± √200) / 2
x = (40 ± 14.14) / 2
Using the positive square root, we have:
x = (40 + 14.14) / 2
x = 27.07
Similarly, using the negative square root:
x = (40 - 14.14) / 2
x = 12.93
Now, let's analyze the two possible values of x:
- If x = 27.07, the width of the rectangle would exceed the length, which doesn't make sense in the context of a rectangle.
- If x = 12.93, the width would be less than the length, which is also not logical for a rectangle.
Based on these calculations, it seems that there is no appropriate value of x that satisfies the equation and represents a valid garden with a length and width.
Therefore, it seems there is a mistake in the problem statement, or some additional information is missing. It would be best to double-check the given information or consult the source you obtained it from to determine the correct dimensions of your garden and reassess its size.
so far, so good. The area of such a rectangle is
x(40-x) = 40x - x^2
You want
40x - x^2 >= 350
-x^2 + 40x - 350 >= 0
Now, the roots of this function are
12.9 and 27.1
Since it is a parabola opening down, y is positive for x between the roots. So, for any value of x between 12.9 and 27.1, the garden will be more than 350 ft^2.
Or, you could have examined the equation and noted that the vertex is at x=20, where there is maximum area. f(20) = 400, more than big enough.
Just for future reference, the maximum area possible for a rectangle of given perimeter is when the rectangle is a square. So, with 80 feet of fencing, the maximum area is 400 ft^2, in a 20x20 square. So, it should have been immediately clear that there was enough fencing to enclose 350 ft^2.