Answer:
The dimensions that match Alice's rectangle are 7 cm and 8 cm.
Step-by-step explanation:
The perimeter of the rectangle is given by:
[tex] P = 2(a + b) = 30 cm [/tex] (1)
Where a and b are the sides of the rectangle
Also, the area of the rectangle is:
[tex] A = a\times b = 56 cm^{2} [/tex] (2)
By solving equation (1) for a, we have:
[tex] a = 15 cm - b [/tex] (3)
Now, by entering equation (3) into (2) we can find one side of the rectangle:
[tex] 56 cm^{2} = (15 cm - b)\times b [/tex]
[tex] b^{2} - 15b + 56 = 0 [/tex]
Solving the above quadratic equation we have:
b₁ = 7 cm and b₂ = 8 cm
Now, the other side of the rectangle can be calculated with equation (3):
[tex] a = 15 cm - b_{1} = 15 - 7 cm = 8 cm [/tex]
or
[tex] a = 15 cm - b_{2} = 15 - 8 cm = 7 cm [/tex]
Taking the first solution (b₁) or the second (b₂), we find that the dimensions that match Alice's rectangle are 7 cm and 8 cm.
Therefore, the dimensions that match Alice's rectangle are 7 cm and 8 cm.
I hope it helps you!
