The diagonals of a rectangle is 8m longer than it's shorter side. If the area is 60m.sqr., what are its dimentions?
Show the solution.
short side --- x
diagonal --- x+8
other side --- y
x^2 + y^2 = (x+8)^2
x^2 + y^2 = x^2 + 16x + 64
y^2 = 16x + 64
y = √(16x+64)
area = xy = 60
x√(16x+64) = 60
square both sides
x^2(16x+64) = 3600
16x^3 + 64x^2=3600
x^3 + 4x^2 - 225 = 0
looks like x = 5 works
then y = √(16(5) + 64) = 12
the rectangle is 5 m by 12 m
If we had known the numbers came out that "nice" we could have done this by trial and error
e.g. factor of 60:
1x60 , 2x30, 3x20, 4x16, 5x12, 6x10
the only one which give yield a whole-number hypotenuse is
5x12 --->5^2 + 12^2 = 13^2 , and 13 is 8 more than 5
How did you find 13 as c?(a²+b²=c²)
The stadium has a form of a rectangle
salamat😊
When the sides are increased by 1,2 and 3 units respectively , the volume is increased by 25 units
To solve this problem, let's assume the shorter side of the rectangle has a length of "a" meters.
According to the given information, the diagonals of the rectangle are 8 meters longer than its shorter side. Therefore, the length of the longer diagonal is "a + 8" meters.
We know that the diagonals of a rectangle divide it into four congruent right triangles. Using the Pythagorean theorem, we can find the relationship between the sides of these right triangles.
The Pythagorean theorem states that for any right triangle, the square of the hypotenuse (the longest side) is equal to the sum of the squares of the other two sides.
In this case, we have a right triangle with sides "a", "a + 8", and the hypotenuse being the longer diagonal.
Applying the Pythagorean theorem to this right triangle, we have:
(a + 8)^2 = a^2 + a^2
Simplifying this equation, we get:
a^2 + 16a + 64 = 2a^2
Rearranging the equation, we have:
2a^2 - a^2 - 16a - 64 = 0
a^2 - 16a - 64 = 0
Now, we can solve this quadratic equation using factoring, completing the square, or using the quadratic formula. In this case, let's use factoring.
Factoring this equation, we can rewrite it as:
(a - 8)(a + 8) = 0
Setting each factor equal to zero, we have:
a - 8 = 0 or a + 8 = 0
If a - 8 = 0, then a = 8.
If a + 8 = 0, then a = -8.
Since the length of a side cannot be negative, we can ignore the solution a = -8.
Therefore, the shorter side of the rectangle is a = 8 meters.
Now, we can find the length of the longer diagonal by adding 8 to the shorter side:
Length of longer diagonal = a + 8 = 8 + 8 = 16 meters.
We also know that the area of a rectangle is determined by multiplying its length and width. In this case, the area is given as 60 square meters.
Area = length x width
60 = 8 x width (Substituting the given values)
Simplifying the equation, we get:
width = 60 / 8
width = 7.5 meters.
Therefore, the dimensions of the rectangle are 8 meters (shorter side), 7.5 meters (longer side), and 16 meters (longer diagonal).