A rectangular flower bed is to be created so that the length is 3 m more than the width. If the length of the diagonal of the flower bed is 15 m, determine the length and the width.
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
width--- x
length ---- x+3
x^ + (x+3)^2 = 15^2
solve for x, reject the negative answer for x
yes
Oh, a flower bed! How delightful! Let's solve this puzzle, shall we?
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!
To solve this problem, we can use the Pythagorean theorem, which states that in a right-angled triangle, the square of the length of the hypotenuse (diagonal) is equal to the sum of the squares of the other two sides.
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.