You work for a fencing company. A customer called, wanting to fence in his 1,320 square-foot rectangular garden. He ordered 148 feet of fencing, but you forgot to ask him for the width and length of the garden (these dimensions will determine some of the details of the order, so you need the information). You don't want the customer to think that you're incompetent, so you figure out the length and width from the information the customer has already given you. What are the dimensions? Hint, what is the area of the garden, and what is the perimeter of the garden? If you are having trouble using algebra to solve this problem, you may be able to get the answer through guess and check.

Question

You work for a fencing company. A customer called, wanting to fence in his 1,320 square-foot rectangular garden. He ordered 148 feet of fencing, but you forgot to ask him for the width and length of the garden (these dimensions will determine some of the details of the order, so you need the information). You don't want the customer to think that you're incompetent, so you figure out the length and width from the information the customer has already given you. What are the dimensions? Hint, what is the area of the garden, and what is the perimeter of the garden? If you are having trouble using algebra to solve this problem, you may be able to get the answer through guess and check.

Answer 1

perimeter = 2 x + 2 y

where x is width and y is length
so 2 x + 2 y = 148
x + y = 74
also we know area = x*y = 1320
so
y = 1320/x
so
x + 1320/x = 74
x^2 + 1320 = 74 x
x^2 - 74 x = -1320
add (74/2)^2 = 37^2 =1369 to both sides
(x-37)^2 = 49
x = 37 +/- 7
x = 30 or x = 44
well, I think this garden is 30 by 44 feet

Answer 2

Thanks :) It is 30 by 44 feet :)

Answer 3

To determine the dimensions of the garden, we need to use the information given - the area of the garden and the amount of fencing required.

We know that the customer wants to fence in a rectangular garden with an area of 1,320 square feet. Let's call the width of the garden "w" and the length "l".

The formula for the area of a rectangle is A = l * w. In this case, the area is 1,320 square feet.

So we have the equation: 1,320 = l * w.

The customer also ordered 148 feet of fencing, which would be used to enclose the perimeter of the garden. The formula for the perimeter of a rectangle is P = 2l + 2w.

We know the perimeter is 148 feet, so we have the equation: 148 = 2l + 2w.

To solve this system of equations, we can use algebra or guess and check.

Let's try guessing and checking with different values for the length and width until we find a combination that satisfies both equations.

Starting with a guess:
Assume the width is w = 20 feet.

Using the first equation: 1,320 = l * 20,
We can solve for l: l = 1,320 / 20 = 66 feet.

Now, let's plug these values into the second equation: 148 = 2*66 + 2*20.
This simplifies to: 148 = 132 + 40, which is not true.

Let's try another guess:
Assume the width is w = 10 feet.

Using the first equation: 1,320 = l * 10,
We can solve for l: l = 1,320 / 10 = 132 feet.

Now, let's plug these values into the second equation: 148 = 2*132 + 2*10.
This simplifies to: 148 = 264 + 20, which is not true either.

Let's try one more guess:
Assume the width is w = 30 feet.

Using the first equation: 1,320 = l * 30,
We can solve for l: l = 1320 / 30 = 44 feet.

Now, let's plug these values into the second equation: 148 = 2*44 + 2*30.
This simplifies to: 148 = 88 + 60, which is true.

Therefore, the width of the garden is 30 feet, and the length is 44 feet.

Answer 4

To determine the dimensions of the garden, we'll use the given information about the area and perimeter.

The area of a rectangle is calculated by multiplying its length and width. Given that the garden area is 1,320 square feet, we have the equation:

Area = Length x Width = 1,320 (equation 1)

The perimeter of a rectangle is calculated by adding up all the sides. Since the fencing ordered is 148 feet long, it will be used to enclose the entire garden perimeter. This means the sum of the sides will be equal to 148 feet. For a rectangle, the perimeter is given by the equation:

Perimeter = 2 x (Length + Width) = 148 (equation 2)

From equations 1 and 2, we have a system of two equations with two variables (Length and Width). We can solve this system to find the values for the dimensions of the garden.

One way to solve this system algebraically is to solve equation 2 for either Length or Width and substitute that expression into equation 1. However, since the problem also suggests using guess and check, let's utilize that approach.

We know that the length and width of the garden should be positive numbers. We can start by guessing values for the length or width and then calculate the remaining dimension using equation 1.

Since the area is relatively large (1,320 square feet) and the perimeter is comparatively small (148 feet), we can guess that the garden is relatively narrow and long.

Let's start with a guess that the width is 10 feet.

Substituting this width value into equation 1:

Length = Area / Width = 1,320 / 10 = 132 feet

To verify if our guess is correct, we calculate the perimeter using this length and width:

Perimeter = 2 x (Length + Width) = 2 x (132 + 10) = 284 feet

Based on our guess, the perimeter is larger than the given length of fencing (148 feet), which means our width guess of 10 feet is incorrect.

Now, let's try another guess. This time, let's assume the width is 20 feet.

Substituting this width value into equation 1:

Length = Area / Width = 1,320 / 20 = 66 feet

Calculating the perimeter using this length and width:

Perimeter = 2 x (Length + Width) = 2 x (66 + 20) = 172 feet

Based on this guess, the perimeter is still larger than the given length of fencing (148 feet), so our width guess of 20 feet is also incorrect.

We can continue this process of guess and check until we find the correct dimensions that satisfy both the area and perimeter conditions.