The length of a rectangle is 3 m less than the diagonal and the width is 8 m less than the diagonal. If the area is 84 m2, how long is the diagonal? Round your answer to the nearest tenth.
L^2 + w^2 = d^2
(d-3)^2 + (d-8)^2 = d^2
d^2 - 6 d + 9 + d^2 -16 d + 64 = d^2
d^2 - 22 d + 73 = 0
d = [ 22 +/- sqrt(484 - 292) ] / 2
d = [ 22 +/- 13.9 ] / 2
d = 17.9 or 8.1
8.1 is impossible if the area is 84 so d = 17.9
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check
then w = about 10 and L = about 15
so area = 150 m^2
beats me
To find the length of the diagonal, we need to use the formula for the area of a rectangle which is length multiplied by width:
Area = length * width
Given that the area is 84 m², we have:
84 = length * width
We also know that the length is 3 m less than the diagonal, and the width is 8 m less than the diagonal. Let's represent the diagonal as "d".
So we have two equations based on the information given:
length = d - 3
width = d - 8
Substituting these values into the area equation, we get:
84 = (d - 3) * (d - 8)
Now, we can solve this quadratic equation to find the value of "d", the length of the diagonal. We will simplify the equation and solve for "d".
Expanding the equation:
84 = d² - 11d + 24
Rearranging the equation:
d² - 11d - 60 = 0
Now we have a quadratic equation in standard form: ax² + bx + c = 0, where a = 1, b = -11, and c = -60.
To solve this quadratic equation, we can either factor it or use the quadratic formula. Since factoring is not straightforward in this case, let's use the quadratic formula:
d = (-b ± √(b² - 4ac)) / (2a)
Substituting the values:
d = (-(-11) ± √((-11)² - 4 * 1 * (-60))) / (2 * 1)
Simplifying:
d = (11 ± √(121 + 240)) / 2
d = (11 ± √361) / 2
d = (11 ± 19) / 2
So the two possible values for d are:
d₁ = (11 + 19) / 2 = 30 / 2 = 15
d₂ = (11 - 19) / 2 = -8 / 2 = -4
Since the length of a diagonal cannot be negative, we disregard d₂.
Therefore, the length of the diagonal (rounded to the nearest tenth) is approximately 15 meters.