The length of a rectangle is 2 inches more than the width. If the diagonal is 20−−√ inches, find the area of the rectangle.
Responses
A 4 in24 in 2
B 8 in28 in 2
C 16 in216 in 2
D 32 in2
Let's assume the width of the rectangle is x inches.
According to the problem, the length of the rectangle is 2 inches more than the width, so the length would be x + 2 inches.
We can use the Pythagorean theorem to find the length of the diagonal:
(x + 2)^2 + x^2 = (20 - √2)^2
Simplifying the equation, we get:
2x^2 + 4x + 4 = 416 - 40√2 + 2
2x^2 + 4x - 414 = -40√2
We can ignore the negative solution since length cannot be negative. So we have:
2x^2 + 4x - 414 = 40√2
x^2 + 2x - 207 = 20√2
To solve this equation, we can use the quadratic formula:
x = (-2 ± √(2^2 - 4(1)(-207))) / 2(1)
x = (-2 ± √(4 + 828)) / 2
x = (-2 ± √(832)) / 2
x = (-2 ± 4√13) / 2
x = -1 ± 2√13
Since x = width, we can ignore the negative solution. Therefore, the width is -1 + 2√13 inches.
The length would be x + 2, so the length is -1 + 2√13 + 2 = 1 + 2√13 inches.
The area of the rectangle is given by the formula: Area = length × width
Area = (1 + 2√13)(-1 + 2√13)
Area = -1 + 2√13 + 2√13 - 4(√13)^2
Area = -1 + 4√13 - 4(13)
Area = -1 + 4√13 - 52
Area = -53 + 4√13
Therefore, the area of the rectangle is -53 + 4√13 square inches.
Let's assume the width of the rectangle is "x" inches.
According to the given information, the length of the rectangle is 2 inches more than the width. So, the length can be represented as "x + 2" inches.
We can use the Pythagorean Theorem to find the diagonal of the rectangle. The formula is: diagonal^2 = length^2 + width^2.
Substituting the values, we can write: (20 - √diagonal)^2 = (x + 2)^2 + x^2.
Expanding the equation, we get: 400 - 40√diagonal + diagonal = x^2 + 4x + 4 + x^2.
Simplifying the equation further, we have: 2x^2 + 4x - 40√diagonal + diagonal - 396 = 0.
Since we know the value of the diagonal is 20 - √diagonal, we can substitute it into the equation: 2x^2 + 4x - 40(20 - √diagonal) + (20 - √diagonal) - 396 = 0.
Simplifying and rearranging the equation, we have: 2x^2 + 4x - 792 + √diagonal(39) = 0.
Now, we can solve this quadratic equation using the quadratic formula: x = (-b ± √(b^2 - 4ac)) / (2a).
Substituting the values, we get: x = (-4 ± √(4^2 - 4(2)(-792 + √diagonal(39)))) / (2(2)).
Simplifying further, we have: x = (-4 ± √(16 + 3168 - 1584√diagonal(39))) / 4.
After simplifying the equation, we have: x = (-1 ± √(1 + 792 - 396√diagonal(39))) / 2.
Now, we have two possible values for the width of the rectangle. Let's calculate them separately.
1. For x = (-1 + √(1 + 792 - 396√diagonal(39))) / 2:
Calculate the value of x using the given formula.
2. For x = (-1 - √(1 + 792 - 396√diagonal(39))) / 2:
Calculate the value of x using the given formula.
Once we have the width value, we can find the length by adding 2 inches to it.
Finally, we can calculate the area of the rectangle by multiplying the width and the length.
The answer will be one of the given options: A) 4 in^2, B) 8 in^2, C) 16 in^2, D) 32 in^2.