the width of a rectangle is 9 less than twice the length. if the area os the rectangle is 41cm^2 what is the length of the diagonal?
if x is the length, 2x-9 = width
x(2x-9) = 41
2x^2 - 9x - 41 = 0
x = 1/4 (9±√409)
since we want a positive length,
length = 1/4 (9+√409)
width = 1/2 (9+√409)-9 = (√409 - 9)/2
diagonal d^2 = 1/16 ((9+√409)^2 + 4(√409-9)^2)
= 1/8 (1225-27√409)
d = √(that)
that is an unusual answer. I suspect a typo somewhere. If so, make the adjustment and redo the calculations.
The width of a rectangle is 9 less than twice its length. If the area of the rectangle is 55 cm22, what is the length of the diagonal?
To find the length of the diagonal of a rectangle, we first need to determine the length and width of the rectangle.
Let's assume the length of the rectangle is "l" cm. According to the problem, the width of the rectangle is 9 less than twice the length. So, the width can be represented as (2l - 9) cm.
Given that the area of the rectangle is 41 cm^2, we can use the following formula to find the area:
Area = Length × Width
Substituting the values from the problem, we have:
41 = l × (2l - 9)
Now, let's solve this equation to find the length of the rectangle:
Step 1: Distribute the length on the right side of the equation:
41 = 2l^2 - 9l
Step 2: Rearrange the equation to form a quadratic equation:
2l^2 - 9l - 41 = 0
Step 3: This quadratic equation does not factor easily, so we can use the quadratic formula to find the value of "l":
l = (-b ± √(b^2 - 4ac)) / (2a)
In this case, a = 2, b = -9, and c = -41:
l = (-(-9) ± √((-9)^2 - 4(2)(-41))) / (2(2))
l = (9 ± √(81 + 328)) / 4
l = (9 ± √(409)) / 4
Now, we have two possible solutions for "l". Let's solve for both:
l₁ = (9 + √409) / 4
l₂ = (9 - √409) / 4
Step 4: Since the length of a rectangle cannot be negative, we can discard the negative solution (l₂). Thus, we have:
l = (9 + √409) / 4
Now that we have the length of the rectangle, we can find the diagonal length using the Pythagorean Theorem.
According to the theorem, the square of the diagonal (d^2) is equal to the sum of the squares of the length (l^2) and the width (w^2) of the rectangle:
d^2 = l^2 + w^2
Substituting the values:
d^2 = l^2 + (2l - 9)^2
Calculating this equation, we get:
d^2 = [(9 + √409) / 4]^2 + [2(9 + √409) / 4 - 9]^2
Simplifying the equation further will provide the value of the diagonal (d).