A small painting has an area of 400 c m
2
. The length is 4 more
than 2 times the width. Find the dimensions of the painting. Solve
by completing the square. Round answers to the nearest tenth of
a centimeter.
width --- x
length -- 2x+4
x(2x + 4) = 400
2x^2 + 4x = 400
x^2 + 2x = 200
x^2 + 2x + 1 = 200 + 1
(x+1)^2 = 201
x+1 = ± √201
x = -1± √201
x = appr 13.18 cm or a negative, which we will reject
The width is appr 13.2 and the length is appr 30.4
no u
Oh, I love solving art-related problems! Let's find those dimensions with a touch of humor, shall we?
Given that the area of the painting is 400 cm^2, we can say:
Let's assume the width of the painting is x cm. So, the length will be (2x + 4) cm.
Now, to solve it using completing the square, let's start by setting up the equation:
x * (2x + 4) = 400
Expanding the equation, we get:
2x^2 + 4x = 400
Next, let's move everything to one side to make it easier to complete the square:
2x^2 + 4x - 400 = 0
Now, let's complete the square like a master artist:
To complete the square, we need to take the coefficient of x (4) and divide it by 2, giving us 2. Then we square it, giving us 4.
Adding 4 to both sides, we get:
2x^2 + 4x + 4 - 400 = 4
Simplifying:
2(x^2 + 2x + 1) = 4 + 400
Now, let's have some fun with factorization:
2(x + 1)(x + 1) = 404
Dividing both sides by 2:
(x + 1)(x + 1) = 202
Taking the square root of both sides:
x + 1 = ±√202
Simplifying:
x = -1 ± √202
Since we can't have a negative width, we'll discard the negative value.
So, by rounding to the nearest tenth, the width of the painting is approximately √202 - 1 cm.
Now, using the equation for the length, which is (2x + 4), we can find it as follows:
Length = 2(√202 - 1) + 4
Calculating that, we get:
Length ≈ 2√202 + 2
And that's it! The dimensions of the painting, rounded to the nearest tenth, are approximately:
Width ≈ √202 - 1 cm
Length ≈ 2√202 + 2 cm
Keep on creating art with math, my friend!
To find the dimensions of the painting, let's assume the width of the painting is x cm.
According to the problem, the length is 4 more than 2 times the width. Therefore, the length can be expressed as 2x + 4 cm.
The area of the painting is given as 400 cm^2. The formula for the area of a rectangle is length × width, so we can create the equation:
x(2x + 4) = 400
Now let's solve this equation by completing the square:
Step 1: Distribute x
2x^2 + 4x = 400
Step 2: Move 400 to the right side
2x^2 + 4x - 400 = 0
Step 3: Divide the entire equation by 2
x^2 + 2x - 200 = 0
Step 4: Move the constant term (-200) to the right side
x^2 + 2x = 200
Step 5: Find a term that can be completed to form a perfect square; in this case, it is (2/2)^2 = 1
x^2 + 2x + 1 = 200 + 1
(x + 1)^2 = 201
Step 6: Take the square root of both sides
√(x + 1)^2 = √201
x + 1 = ±√201
Step 7: Subtract 1 from both sides
x = -1 ±√201
The width, x, cannot be negative in this context, so we take the positive square root:
x = √201 - 1
Now we can find the length by substituting the value of x into the expression for the length:
Length = 2x + 4
Length = 2(√201 - 1) + 4
Calculating this expression, we get:
Length ≈ 2.3 cm
So, the dimensions of the painting are approximately:
Width ≈ √201 - 1 cm
Length ≈ 2.3 cm
To solve this problem using completing the square method, let's first define the variables:
Let's say the width of the painting is 'w' cm.
According to the given information, the length is 4 more than 2 times the width. So, the length can be expressed as (2w + 4) cm.
The area of the painting can be calculated using the formula: Area = length × width.
In this case, the area is given as 400 cm². So, we have the equation:
400 = (2w + 4) × w
To solve this equation by completing the square, let's follow these steps:
Step 1: Expand the equation.
400 = 2w² + 4w
Step 2: Move all terms to one side to form a quadratic equation in standard form.
2w² + 4w - 400 = 0
Step 3: Divide the entire equation by the coefficient of w² to make the coefficient of w² equal to 1.
w² + 2w - 200 = 0
Step 4: To complete the square, take half of the coefficient of 'w' (which is 2) and square it: (2/2)² = 1. Add this value to both sides of the equation.
w² + 2w + 1 - 200 + 1 = 0
Simplifying the equation:
w² + 2w + 1 - 199 = 0
(w + 1)² = 199
Step 5: Take the square root of both sides of the equation.
√[(w + 1)²] = ± √199
w + 1 = ± √199
Step 6: Solve for 'w'.
w = -1 ± √199
Since dimensions cannot be negative, we can disregard the negative value.
Therefore, the width of the painting (rounded to the nearest tenth of a centimeter) is approximately:
w ≈ -1 + √199 ≈ 13.9 cm
Now, we can find the length by substituting this value back into the expression (2w + 4):
length ≈ 2(13.9) + 4 ≈ 31.8 cm
Thus, the dimensions of the painting are approximately:
Width ≈ 13.9 cm
Length ≈ 31.8 cm