A school has a classroom with a length eight feet less than twice the width. If a group of parents purchased 400 square feet of carpet for the room, what is the approximate length of the room?
L=2W-8 L*W=400
(2W-8)W=400
2w^2-8w=400
To solve the equation, we will rearrange it to set it equal to zero:
2w^2 - 8w - 400 = 0
Next, we can solve this quadratic equation by factoring, completing the square, or using the quadratic formula. In this case, let's use factoring.
We can start by factoring out a common factor of 2:
2(w^2 - 4w - 200) = 0
Next, we need to factor the expression within the parentheses:
(w - 20)(w + 10) = 0
Now we can set each factor equal to zero and solve for w:
w - 20 = 0 or w + 10 = 0
For the first equation, adding 20 to both sides gives us:
w = 20
For the second equation, subtracting 10 from both sides gives us:
w = -10
A length cannot be negative, so we can ignore the solution w = -10.
Therefore, the width of the room is 20 feet.
Finally, to find the approximate length, we can substitute the value of the width into the equation L = 2W - 8:
L = 2(20) - 8
L = 40 - 8
L ≈ 32 feet
The approximate length of the room is 32 feet.
To find the approximate length of the room, we need to solve the equation 2w^2 - 8w = 400.
Step 1: Rewrite the equation in standard form, which is a quadratic equation: 2w^2 - 8w - 400 = 0.
Step 2: To solve this quadratic equation, we can use factoring, completing the square, or the quadratic formula. In this case, factoring may be a bit complicated, so let's use the quadratic formula: w = (-b ± √(b^2 - 4ac)) / 2a.
For our equation, a = 2, b = -8, and c = -400.
Substitute these values into the quadratic formula: w = (-(-8) ± √((-8)^2 - 4(2)(-400))) / (2(2)).
Simplify within the square root: w = (8 ± √(64 + 3200)) / 4.
Further simplification: w = (8 ± √(3264)) / 4.
Step 3: Evaluate the square root of 3264 approximately: √3264 ≈ 57.08.
Now, substitute this value back into the equation: w = (8 ± 57.08) / 4.
We have two possible solutions: w1 = (8 + 57.08) / 4 = 65.08 / 4 ≈ 16.27 and w2 = (8 - 57.08) / 4 = -49.08 / 4 ≈ -12.27.
Since w represents the width of the classroom, we can ignore the negative value, as it does not make sense in this context.
Therefore, the approximate width of the classroom is about 16.27 feet.
Step 4: Use the value of w to find the approximate length of the classroom using the given formula L = 2w - 8.
L = 2(16.27) - 8.
L = 32.54 - 8.
L ≈ 24.54.
So, the approximate length of the room is about 24.54 feet.