If the diagonal of a rectangle is 25 cm and the length is 3cm more than triple the width what are the dimensions
625 = w^2 + (3w+3)^2 = w^2 + 9(w+1)^2 = w^2+9w^2+18w+9
625 = 10 w^2 + 18 w + 9
10 w^2 + 18 w - 616 = 0
5 w^2 + 9 w - 308 = 0
https://www.mathsisfun.com/quadratic-equation-solver.html
w = 7 (or - 8.8 :)
L = 21 + 3
width --- x
length -- 3x + 3
x^2 + (3x+3)^2 = 25^2
x^2 + 9x^2 + 18x + 9 - 625 = 0
10x^2 + 18x - 616 = 0
solve the quadratic using your favourite method
make sure to reject the negative and only use the positive answer
recognize the familiar 7-24-25 right triangle?
To find the dimensions of the rectangle, we can set up a system of equations.
Let's assume the width of the rectangle is x cm.
According to the given information, the length of the rectangle is 3 cm more than triple the width. Therefore, the length will be (3x + 3) cm.
We also know that the diagonal of the rectangle is 25 cm.
Using the Pythagorean theorem, we can relate the length, width, and diagonal of the rectangle:
(diagonal)^2 = (length)^2 + (width)^2
Replacing the variables with the given values, we have:
25^2 = (3x + 3)^2 + x^2
Simplifying this equation, we get:
625 = 9x^2 + 18x + 9 + x^2
Combine like terms:
625 = 10x^2 + 18x + 9
Rearranging and setting the equation to zero:
10x^2 + 18x - 616 = 0
To solve this quadratic equation, we can use factoring, completing the square, or the quadratic formula. In this case, let's use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
For our equation, a = 10, b = 18, and c = -616. Substituting these values into the quadratic formula, we get:
x = (-18 ± √(18^2 - 4*10*(-616))) / (2*10)
Simplifying further, we have:
x = (-18 ± √(324 + 24640)) / 20
x = (-18 ± √24964) / 20
Now, we can calculate the two possible values of x:
x1 = (-18 + √24964) / 20
x2 = (-18 - √24964) / 20
Evaluating these equations, we obtain:
x1 ≈ 4.57 cm
x2 ≈ -13.57 cm
Since width cannot be negative, we discard the negative value.
Therefore, the width of the rectangle is approximately 4.57 cm.
Now, we can find the length using the earlier expression:
Length = 3x + 3
Length ≈ 3(4.57) + 3
Length ≈ 13.71 + 3
Length ≈ 16.71 cm
Hence, the dimensions of the rectangle are approximately 4.57 cm by 16.71 cm.