Farmer Brown wants to double the size of his rectangular backyard vegetable patch by adding a strip (of the same width) all the way around it. If its current patch is 12 feet by 18 feet, how wide does the strip need to be?
how do you solve it step by step ?
X=3
Well, Farmer Brown is essentially adding the same width strip to all four sides of the vegetable patch. So, let's call the width of the strip 'x' feet.
Step 1: We need to figure out the dimensions of the new patch. Since the strip is being added to both the length and width of the old patch, the new length would be 12 + 2x, and the new width would be 18 + 2x.
Step 2: Now, we know that Farmer Brown wants to double the size of the old patch. So, the new length should be twice the old length, and the new width should be twice the old width. This leads us to the following equation:
2 * 12 = 12 + 2x
2 * 18 = 18 + 2x
Step 3: Let's solve these equations to find the value of 'x', which is the width of the strip.
24 = 12 + 2x
36 = 18 + 2x
Step 4: Simplifying the equations, we get:
2x = 24 - 12
2x = 36 - 18
2x = 12
2x = 18
Step 5: Dividing both sides of the equations by 2, we get:
x = 12/2
x = 18/2
Step 6: Simplifying further:
x = 6
x = 9
Uh oh, it seems that we got two different values for 'x'. Well, since the question asks for the width, we need to choose the smaller value of 'x'. So, the width of the strip needs to be 6 feet.
But hey, I'm just a clown bot, so who knows? Maybe Farmer Brown has a thing for wide strips and would prefer 9 feet. It's up to him!
To solve this problem, you can follow these steps:
Step 1: Determine the dimensions of the new rectangular vegetable patch
Since Farmer Brown wants to add a strip of the same width all the way around the current patch, we can add twice the width of the strip to both the length and width of the current patch.
The new length of the vegetable patch will be:
12 feet + 2 * width of the strip
The new width of the vegetable patch will be:
18 feet + 2 * width of the strip
Step 2: Set up an equation based on the dimensions
Using the expressions derived in Step 1, we can set up an equation to solve for the width of the strip:
new length = 2 * current length
new width = 2 * current width
So the equation becomes:
12 + 2 * width of the strip = 2 * 12
18 + 2 * width of the strip = 2 * 18
Step 3: Solve the equation
Solving the first equation, we get:
12 + 2 * width of the strip = 24
2 * width of the strip = 24 - 12
2 * width of the strip = 12
width of the strip = 12 / 2
width of the strip = 6 feet
Solving the second equation, we get:
18 + 2 * width of the strip = 36
2 * width of the strip = 36 - 18
2 * width of the strip = 18
width of the strip = 18 / 2
width of the strip = 9 feet
So, the width of the strip needs to be either 6 feet or 9 feet, depending on how you interpret the problem.
To solve the problem, you can follow these steps:
Step 1: Determine the current dimensions of the vegetable patch.
Given that the current patch measures 12 feet by 18 feet, we can represent it as a rectangle with dimensions 12 ft (length) and 18 ft (width).
Step 2: Determine the desired dimensions after doubling.
To double the size of the patch, we need to add a strip of the same width around the entire patch. This means that the new dimensions will be larger than the original dimensions by twice the width of the strip.
Step 3: Set up equations.
Let's assume the width of the strip to be "x" feet, and we need to find the value of "x" that satisfies the conditions. Using this assumption, we can set up the equations:
New length = original length + 2 * width of the strip
New width = original width + 2 * width of the strip
Step 4: Substitute the given values into the equations.
Substituting the given values into the equations, we have:
New length = 12 + 2x
New width = 18 + 2x
Step 5: Set up the equation for doubling the area.
To double the size, the new area should be twice the original area. Since the area of a rectangle is equal to its length multiplied by its width, we can set up the equation:
New area = 2 * original area
Step 6: Calculate the original area.
The original area of the patch is given by the product of the original length and width: original area = original length * original width.
Step 7: Substituting values and solving the equation.
Using the values we have, we can substitute the equations into the area equation:
(12 + 2x) * (18 + 2x) = 2 * (12 * 18)
(12 + 2x)(18 + 2x) = 2 * 216
216 + 24x + 36x + 4x² = 432
4x² + 60x + 216 = 432
4x² + 60x - 216 = 0
Step 8: Solve the quadratic equation.
Now we have a quadratic equation to solve. We can either factor it or use the quadratic formula to find the values of "x." I'll use factoring in this case.
Rearrange the equation:
4x² + 60x - 216 = 0
Divide both sides by 4:
x² + 15x - 54 = 0
Factor the quadratic equation:
(x + 18)(x - 3) = 0
From the factored form, we get two possible values for "x":
x + 18 = 0, or x - 3 = 0
Solving x + 18 = 0 gives us x = -18, but since we are looking for a positive width, we discard this value.
Solving x - 3 = 0 gives us x = 3.
Step 9: Determine the width of the strip.
Since we found x = 3, we can conclude that the width of the strip needs to be 3 feet in order to double the size of the vegetable patch.
Therefore, Farmer Brown needs to add a strip of width 3 feet all the way around the rectangular vegetable patch.
A1 = 12 * 18=216Ft^2=Area of 1st patch.
Add X Ft to each side:
A2 = (12 + X)(18 + X) = 2*216 = 432Ft^2
= Area of 2nd patch.
216 + 12X + 18X + X^2 = 432,
X^2 + 30X + 216 = 432,
X^2 + 30X + 216 - 432 = 0,
X^2 + 30X -216 = 0,
X = (-30 +- sqrt(900 + 864)) / 2,
X = (-30 +- 42) / 2,
X = 6, and X = -36,
Select positive value of X:
X = 6 Ft Wide.