The diagonal of a rectangle is 8m longer than its shorter side. If the area of the rectangle is 60 square meters. find its dimensions.
length = a
width = b
a b = 60
so b = 60/a
sqrt (a^2 + b^2) = b+8
a^2 + b ^2 = b^2 + 16 b + 64
a^2 - 16 b - 64 = 0
a^2 - 16 (60/a) - 64 = 0
a^3 - 960 - 64 a = 0
a^3 - 64 a - 960 = 0
try a = 11. a = 11 a = 12
if a = 12
1728 - 768 = 960 !!!!
so a = 12
then b = 60/12 = 5
Well, let's call the shorter side of the rectangle "x". So according to the information given, the diagonal would be "x + 8" meters long. Now, we can use the Pythagorean theorem to find the longer side. Since the diagonal, shorter side, and longer side form a right triangle, we can use the formula:
(diagonal)^2 = (shorter side)^2 + (longer side)^2
Substituting the values:
(x + 8)^2 = x^2 + (longer side)^2
Expanding and simplifying:
x^2 + 16x + 64 = x^2 + (longer side)^2
Canceling out x^2 on both sides:
16x + 64 = (longer side)^2
Now, we know that the area of the rectangle is equal to the product of two adjacent sides. In this case, it's equal to x*(longer side) = 60. Rearranging the equation:
(longer side) = 60 / x
Substituting this value back into our previous equation:
16x + 64 = (60 / x)^2
Simplifying further:
16x + 64 = 3600 / x^2
Cross-multiplying:
16x^3 + 64x^2 = 3600
Dividing by 16:
x^3 + 4x^2 = 225
Now, I'm afraid my calculations have gotten a bit tangled up, and I can't untangle them from this clown wig. So, why don't you give it a go? Remember to take a deep breath, double-check your work, and you'll find the dimensions of that rectangle in no time!
Let's assume that the shorter side of the rectangle measures x meters.
According to the given information, the diagonal of the rectangle is 8 meters longer than the shorter side. This means that the length of the diagonal can be expressed as x + 8 meters.
We can use the Pythagorean theorem to determine the relationship between the sides of a right-angled triangle within the rectangle.
The Pythagorean theorem states that in a right-angled triangle, the square of the length of the hypotenuse (the diagonal in this case) is equal to the sum of the squares of the other two sides.
Using this information, we can set up the following equation:
x^2 + (x + 8)^2 = diagonal^2
Expanding the equation gives:
x^2 + x^2 + 16x + 64 = diagonal^2
2x^2 + 16x + 64 = diagonal^2
We also know that the area of the rectangle is equal to the product of its two sides. In this case, the area is equal to 60 square meters:
x * (x + 8) = 60
x^2 + 8x - 60 = 0
To solve this quadratic equation for x, we can either:
1. Factor the equation:
(x - 4)(x + 12) = 0
Therefore, x can be either 4 or -12. Since the dimension cannot be negative, we discard x = -12.
2. Use the quadratic formula:
x = (-b ± √(b^2 - 4ac))/(2a)
Applying the formula to our quadratic equation, where a = 1, b = 8, and c = -60:
x = (-8 ± √(8^2 - 4(1)(-60)))/(2 * 1)
x = (-8 ± √(64 + 240))/2
x = (-8 ± √304)/2
After simplifying, we have:
x = (-8 ± 16.49)/2
x ≈ -12.25 or x ≈ 4.25
Again, we discard x = -12.25 as negative values are not applicable in this case.
Therefore, the shorter side of the rectangle measures approximately 4.25 meters.
To find the other side, we can use the equation for the area of a rectangle:
length * width = area
Therefore the length of the rectangle is:
length = area/width
= 60/4.25
≈ 14.12 meters.
So, the dimensions of the rectangle are approximately 4.25 meters by 14.12 meters.
To find the dimensions of the rectangle, we can use a systematic approach.
Let's assume the shorter side of the rectangle is x meters.
According to the given information, the diagonal of the rectangle is 8m longer than its shorter side. So, the length of the diagonal can be represented as x + 8.
We know that the area of a rectangle is equal to its length multiplied by its width. In this case, the area is given as 60 square meters.
So, the equation representing the area is:
Area = Length × Width
Substituting the values, we get:
60 = (x + 8) × x
Expanding the equation, we have:
60 = x^2 + 8x
Rearranging the equation to the standard quadratic form, we get:
x^2 + 8x - 60 = 0
Now, we can solve this quadratic equation to find the value of x.
Using factoring, we can write the equation as:
(x - 4)(x + 12) = 0
Setting each factor equal to zero, we have:
x - 4 = 0 or x + 12 = 0
Solving each equation, we find that:
x = 4 or x = -12
Since the length cannot be negative, we discard x = -12 as a valid solution.
Therefore, the shorter side of the rectangle is x = 4 meters.
Now, we can find the other dimension by substituting the value of x into the equation:
Length = x + 8
Length = 4 + 8
Length = 12 meters
So, the dimensions of the rectangle are 4 meters by 12 meters.