The length of a rectangle is 5 ft greater than 4 times its width. The area is 369 sq ft . What is the length?
Let's solve this step-by-step.
Step 1: Assign variables
Let's assign the width of the rectangle as 'w'.
Step 2: Find the length
According to the problem, the length of the rectangle is 5 ft greater than 4 times its width. So, the length can be expressed as 4w + 5.
Step 3: Use the formula for the area of a rectangle
The formula for the area of a rectangle is length * width. In this case, the area is given as 369 sq ft, so we can write the equation as:
(4w + 5) * w = 369
Step 4: Solve the equation
Expand the equation:
4w^2 + 5w = 369
Rearrange the equation by bringing all terms to one side:
4w^2 + 5w - 369 = 0
Now, we can solve this quadratic equation to find the value of 'w'.
Step 5: Use the quadratic formula
The quadratic formula is given by:
w = (-b ± √(b^2 - 4ac)) / (2a)
For our equation, a = 4, b = 5, and c = -369. Substituting these values, we get:
w = (-5 ± √(5^2 - 4 * 4 * -369)) / (2 * 4)
w = (-5 ± √(25 + 5912)) / 8
w = (-5 ± √5937) / 8
Step 6: Calculate the square root
Using a calculator, we find that √5937 is approximately equal to 77.08.
Step 7: Solve for 'w'
Now, plug in the values of √5937 and solve for 'w':
w = (-5 + 77.08) / 8 ≈ 9.88
w = (-5 - 77.08) / 8 ≈ -10.95
We can ignore the negative value since width cannot be negative.
So, the width of the rectangle is approximately 9.88 ft.
Step 8: Find the length
Using the length formula, substitute the value of 'w':
Length = 4w + 5
Length = 4(9.88) + 5
Length ≈ 39.52 + 5
Length ≈ 44.52
Therefore, the length of the rectangle is approximately 44.52 ft.
To find the length of the rectangle, we need to set up an equation using the given information.
Let's denote the width of the rectangle as "w" ft.
According to the given information, the length is 5 ft greater than 4 times the width, which can be expressed as "4w + 5" ft.
The formula for the area of a rectangle is: Area = Length * Width.
In this case, the area is given as 369 sq ft, so we have the equation:
369 = (4w + 5) * w
Now we can solve this equation to find the value of "w" and then calculate the length.
To solve the equation:
1. Distribute the (4w + 5) term: 369 = 4w^2 + 5w
2. Rearrange the equation to form a quadratic equation: 4w^2 + 5w - 369 = 0
3. This quadratic equation can be solved using factoring, completing the square, or the quadratic formula.
However, in this case, factoring may not be straightforward, and completing the square could be a bit complex. Therefore, let's use the quadratic formula to find the value of "w".
The quadratic formula is: w = (-b ± √(b^2 - 4ac)) / 2a
In our equation, a = 4, b = 5, and c = -369.
Using these values in the quadratic formula, we get:
w = (-5 ± √(5^2 - 4 * 4 * -369)) / (2 * 4)
Calculating this expression will give two possible values of "w": w1 and w2. We can solve for both values and check if either of them meets the given conditions.
After finding the values of "w", we can substitute them into the expression "4w + 5" to find the length of the rectangle corresponding to each width value.
L = 4W + 5
L * W = 369
Substitute 4W+5 for L in second equation and solve for W. Insert that value into the first equation and solve for L. Check by inserting both values into the second equation.