The length of a rectangle is 4 m less than the diagonal and the width is 7 m less than the diagonal. If the area is 76 m2, how long is the diagonal? Round your answer to the nearest tenth.
If the sides are x and y, and the diagonal is d,
x = d-7
y = d-4
so,
d = x+7 = y+4
That is, y = x+3
xy = 76, so
x(x+3) = 76
x = 7.35
So, the rectangle is 7.35 x 10.35
area = 76
diagonal is 12.69
To solve this problem, we'll use the formula for the area of a rectangle and the Pythagorean theorem.
Let's start by setting up the equation for the area of a rectangle:
A = length * width
Given that the area is 76 m², we have:
76 = length * width
Next, let's express the length and width in terms of the diagonal.
We are told that the length is 4 m less than the diagonal, so we can say:
length = diagonal - 4
Similarly, the width is 7 m less than the diagonal, so we can say:
width = diagonal - 7
Substituting these expressions into the equation for the area, we have:
76 = (diagonal - 4) * (diagonal - 7)
Expanding the equation, we get:
76 = diagonal² - 11diagonal + 28
Rearranging the terms, we have:
diagonal² - 11diagonal + 28 - 76 = 0
Simplifying further, we have:
diagonal² - 11diagonal - 48 = 0
Now, we can solve this quadratic equation for the diagonal. We can either factor it or use the quadratic formula. Let's use the quadratic formula:
diagonal = (-b ± √(b² - 4ac)) / (2a)
For our equation, a = 1, b = -11, and c = -48. Plugging in these values, we have:
diagonal = (-(-11) ± √((-11)² - 4(1)(-48))) / (2(1))
diagonal = (11 ± √(121 + 192)) / 2
diagonal = (11 ± √313) / 2
Using a calculator to evaluate √313, we find that √313 ≈ 17.7.
Now, let's substitute this value back into our equation to find the two possible values of the diagonal:
diagonal₁ = (11 + 17.7) / 2 ≈ 28.7
diagonal₂ = (11 - 17.7) / 2 ≈ -3.4
Since a diagonal cannot be negative, we can discard diagonal₂.
Therefore, the length of the diagonal (rounded to the nearest tenth) is approximately 28.7 meters.