Solve by completing the square: A rectangular patio has a length of (x + 6) m, a width of (x + 8) m, and a total area of 400 m2. Find the dimensions to the nearest tenth.
Please help me with this math problem. Thanks so much.
(x+6)(x+8) = 400
x^2+14x+48 = 400
x^2+14x+49 = 401
(x+7)^2 = 401
x+7 = ±√401
x = -7±√401
we want a positive value for x+6, so x=-7+√401 ≈13.025
So the dimensions are 19.025 by 21.025
Note that (x+6)(x+8) = (x+7-1)(x+7+1) = (x+7)^2 - 1^2 as seen above
2021 and they’re still giving out the same pieces of homework
m^2*
I need help too
Ah, the joy of completing the square! Let me put on my math hat for a moment. We know that the area of a rectangle is given by the formula: A = length * width. So, in this case, we have the equation: (x + 6)(x + 8) = 400.
Now, let's multiply the two binomials: x^2 + 14x + 48 = 400. To complete the square, we need to make the left side a perfect square trinomial. We can do this by adding (14/2)^2 = 49 to both sides.
The equation becomes: x^2 + 14x + 48 + 49 = 400 + 49. Simplifying, we get: x^2 + 14x + 97 = 449.
Now, we rewrite the left side as a square of a binomial: (x + 7)^2 = 449. Taking the square root of both sides gives us: x + 7 = √449.
Solving for x, we get x = -7 ± √449. Since we can't have negative lengths for our patio, we ignore the negative value.
Therefore, the length is (x + 6) = (-7 + √449 + 6), and the width is (x + 8) = (-7 + √449 + 8).
To the nearest tenth, the dimensions are approximately: Length ≈ 2.7m and Width ≈ 4.7m. Voila!
To solve this problem, we need to use the concept of completing the square.
1. First, we find the equation for the area of the rectangular patio. The area of a rectangle is given by the product of its length and width, so we have:
Area = (x + 6)(x + 8) = 400
2. Next, we expand the equation:
x^2 + 6x + 8x + 48 = 400
3. Combine like terms:
x^2 + 14x + 48 = 400
4. Move the constant term to the other side of the equation:
x^2 + 14x - 352 = 0
5. To complete the square, we want to create a perfect square trinomial on the left side of the equation. To do this, we need to take half of the coefficient of x, square it, and add it to both sides of the equation. Half of 14 is 7, and squaring it gives us 49. Add 49 to both sides of the equation:
x^2 + 14x + 49 - 352 + 49 = 0 + 49
6. Simplify the equation:
(x + 7)^2 - 303 = 0
7. Rewrite the equation in vertex form by rearranging the terms:
(x + 7)^2 = 303
8. Take the square root of both sides of the equation:
x + 7 = ± sqrt(303)
9. Solve for x by subtracting 7 from both sides of the equation:
x = -7 ± sqrt(303)
Therefore, the possible solutions for x are x ≈ -7 + sqrt(303) and x ≈ -7 - sqrt(303).
To find the dimensions of the patio, substitute each value of x into the expressions for length and width:
Length = (x + 6) m
Width = (x + 8) m
Substituting x = -7 + sqrt(303):
Length = (-7 + sqrt(303) + 6) ≈ -1.3 m
Width = (-7 + sqrt(303) + 8) ≈ 0.7 m
Substituting x = -7 - sqrt(303):
Length = (-7 - sqrt(303) + 6) ≈ -13.3 m
Width = (-7 - sqrt(303) + 8) ≈ -11.3 m
Since negative values for length and width are not meaningful in this context, the only valid solution is approximately Length ≈ -1.3 m and Width ≈ 0.7 m. Remember to consider only the positive values in the given context.