The length of a rectangle is five less than three times its width. Find the dimensions of the rectangle if its area is 782.
I get 18.84
x(3x-5)=782.
The book says 17 m by 46 m
L = (3w-5)
L w = 782
(3w-5)w = 782
3 w^2 -5 w -782 = 0
(3w+46)(x-17) =0
x = 17 or x = -46/3
use x = 17
then L = 782/17 = 46
To find the dimensions of the rectangle, we can start by setting up an equation based on the given information.
Let's assume that the width of the rectangle is x. Therefore, the length of the rectangle would be 3x - 5, as mentioned in the problem.
The formula for the area of a rectangle is length multiplied by width. So we can set up the equation:
x * (3x - 5) = 782
Now we have a quadratic equation. To solve it, we can expand the equation:
3x^2 - 5x - 782 = 0
To solve this quadratic equation, you can either factorize it or use the quadratic formula. Since factoring is not straightforward in this case, we can use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
In our equation, a = 3, b = -5, and c = -782:
x = (-(-5) ± √((-5)^2 - 4 * 3 * (-782))) / (2 * 3)
Simplifying further:
x = (5 ± √(25 + 9368)) / 6
x = (5 ± √9393) / 6
The square root of 9393 is approximately 96.88.
So we have two possible values for x:
x₁ = (5 + 96.88) / 6 ≈ 17.98
x₂ = (5 - 96.88) / 6 ≈ -15.98
Since dimensions cannot be negative, we can discard the negative solution. Therefore, the width is approximately 17.98.
Now, to find the length, substitute the value of x into the expression 3x - 5:
Length = 3 * 17.98 - 5 ≈ 46.94
So, the dimensions of the rectangle are approximately 17.98 m (width) by 46.94 m (length).
Note: The values in the book may vary slightly due to rounding errors or a different method of solving the quadratic equation.
To find the dimensions of the rectangle, we can set up the equation using the given information.
Let's assume the width of the rectangle is "x" units.
According to the given information, the length of the rectangle is five less than three times its width, which can be written as 3x - 5.
The formula to calculate the area of a rectangle is given by length × width. In this case, the area is 782, so we can set up the equation:
x(3x - 5) = 782
Now, let's solve this equation step-by-step:
1. Distribute the x to both terms inside the parentheses:
3x^2 - 5x = 782
2. Rearrange the equation to make it quadratic:
3x^2 - 5x - 782 = 0
3. Now, we can solve this quadratic equation either by factoring, completing the square, or using the quadratic formula. Since factoring seems a bit complex in this case, let's use the quadratic formula:
x = (-b ± sqrt(b^2 - 4ac)) / (2a)
In our equation, a = 3, b = -5, and c = -782. Plugging these values into the quadratic formula:
x = (-(-5) ± sqrt((-5)^2 - 4 * 3 * -782)) / (2 * 3)
x = (5 ± sqrt(25 + 9368)) / 6
x = (5 ± sqrt(9393)) / 6
4. Calculate the square root:
x = (5 ± 96.93) / 6
x ≈ (5 + 96.93) / 6 ≈ 17.82
x ≈ (5 - 96.93) / 6 ≈ -15.82
Since the width cannot be negative for a physical rectangle, we discard the negative value.
Therefore, the width of the rectangle is approximately 17.82 units.
5. To find the length, substitute the value of x into the equation for the length:
Length = 3x - 5
Length = 3 * 17.82 - 5
Length ≈ 46.46
Hence, the dimensions of the rectangle, rounded to the nearest whole number, are approximately 17 units by 46 units.