the length of a rectangle is 6 cm more than its width. if 3 cm are taken from the length and added to the width, the figure becomes a square with an area of 64sq cm. what are the dimensions of the original figure.
Width = W cm.
Length = W+6 cm.
((W+6)-3)(W+3) = 64 cm^2.
(W+3)(W+3) = 64, W^2+6W+9 = 64, W^2+6W-55 = 0, -55 = -5*11, -5+11 = 6 = B, (W-5)(W+11) = 0,
W-5 = 0, W = 5. W+11 = 0, W = -11.
Original Dimensions: W = 5 cm, L = 5+6 = 11 cm.
To solve this problem, we can break it down into steps:
1. Define variables: Let's assume the width of the original rectangle is "w" cm.
2. Set up equations: As per the given information, the length of the rectangle is 6 cm more than its width. Therefore, the length would be (w + 6) cm.
3. Adjust dimensions after the transformation: According to the problem, 3 cm is taken from the length and added to the width. So, the new width becomes (w + 3) cm, and the new length becomes (w + 6 - 3) cm, which simplifies to (w + 3) cm.
4. Write an equation for the area of the transformed figure: Since the transformed figure is a square with an area of 64 sq cm, the dimensions of the transformed figure are equal. Therefore, we can write the equation:
(w + 3) cm)^2 = 64 sq cm
5. Solve the equation: Expanding (w + 3)^2, we get:
w^2 + 6w + 9 = 64
Simplifying further:
w^2 + 6w - 55 = 0
6. Factorize or use the quadratic formula to solve the equation: Factoring the quadratic equation gives us:
(w - 5)(w + 11) = 0
which implies w = 5 or w = -11. Since the width cannot be negative, we discard w = -11.
7. Calculate the length of the original rectangle: Now that we have the width as 5 cm, we can find the length using the equation:
length = width + 6
length = 5 + 6 = 11 cm
8. Verify the solution: We can check the solution by verifying that the transformed figure becomes a square with an area of 64 sq cm when 3 cm are added to the width and subtracted from the length. In this case, the width is 5 + 3 = 8 cm and the length is 11 - 3 = 8 cm. The area of a square with side length 8 cm is indeed 64 sq cm, which confirms our solution.
Therefore, the original dimensions of the rectangle are width = 5 cm and length = 11 cm.