y = x + 3
y^2 + x^2 = 15^2 = 225
(x+3)^2 + x^2 = 225
x^2 + 6 x + 9 + x^2 = 225
2 x^2 + 6 x - 216 = 0
x^2 + 3 x - 108 = 0
https://www.mathsisfun.com/quadratic-equation-solver.html
y^2 + x^2 = 15^2 = 225
(x+3)^2 + x^2 = 225
x^2 + 6 x + 9 + x^2 = 225
2 x^2 + 6 x - 216 = 0
x^2 + 3 x - 108 = 0
https://www.mathsisfun.com/quadratic-equation-solver.html
length ---- x+3
x^ + (x+3)^2 = 15^2
solve for x, reject the negative answer for x
Let's assume the width of the rectangular flower bed is x meters.
According to the problem statement, the length is 3 meters more than the width. So, the length can be expressed as (x + 3) meters.
Now, let's get a little funky with Pythagoras' theorem. The diagonal of a rectangle, in this case, is also the hypotenuse of a right-angled triangle formed by the width, length, and diagonal.
So, using Pythagoras' theorem, we can write the following equation:
x^2 + (x + 3)^2 = 15^2
Solving this equation, we find that the width (x) is 4 meters and the length (x + 3) is 7 meters.
Voila! The width of the rectangular flower bed is 4 meters, and the length is 7 meters. Now go forth and plant some fabulous flowers!
Let's denote the width of the rectangular flower bed as "w" meters.
Given that the length is 3 meters more than the width, we can express it as "w + 3" meters.
Now, let's use the Pythagorean theorem to relate the length, width, and diagonal:
(width)^2 + (length)^2 = (diagonal)^2
Substituting the given values, we have:
(w)^2 + (w + 3)^2 = (15)^2
Expanding and simplifying the equation, we get:
w^2 + w^2 + 6w + 9 = 225
Combining like terms, we have:
2w^2 + 6w + 9 = 225
Rearranging the equation, we get:
2w^2 + 6w - 216 = 0
Now we can solve this quadratic equation for the value of "w". We can either factorize or use the quadratic formula. Let's use the quadratic formula:
w = (-b ± √(b^2 - 4ac)) / (2a)
In this equation, a = 2, b = 6, and c = -216.
Substituting the values in the quadratic formula, we have:
w = (-6 ± √(6^2 - 4(2)(-216))) / (2(2))
Simplifying further:
w = (-6 ± √(36 + 1728)) / 4
w = (-6 ± √1764) / 4
Taking the square root:
w = (-6 ± 42) / 4
Now, calculating both solutions:
w₁ = (-6 + 42) / 4 = 9
w₂ = (-6 - 42) / 4 = -12
Since the width of the flower bed cannot be negative, we discard the negative value.
Therefore, the width of the flower bed is 9 meters.
Now, we can calculate the length of the flower bed using the expression: length = width + 3
length = 9 + 3 = 12 meters
So, the width of the flower bed is 9 meters, and the length is 12 meters.