Answer:
The dimensions of the rectangle are 8 by 7 centimeters.
Step-by-step explanation:
The length of a rectangle is 13 centimeters less than three times its width. In other words:
[tex]\ell = 3w-13[/tex]
Given that the area of the rectangle is 56 square centimeters, we want to determine its dimensions.
Recall that the area of a rectangle is given by:
[tex]A = w \ell[/tex]
Substitute in known values and equations:
[tex](56)=w(3w-13)[/tex]
Solve for w. Distribute:
[tex]3w^2-13w=56[/tex]
Isolate the equation:
[tex]3w^2-13w-56=0[/tex]
Factor. We want to find two numbers that multiply to 3(-56) = -168 and that add to -13.
-21 and 8 suffice. Hence:
[tex]3w^2 - 21w + 8w - 56 = 0 \\ \\ 3w(w-7) + 8(w-7) = 0 \\ \\ (3w+8)(w-7) = 0[/tex]
Zero Product Property:
[tex]3w+8=0\text{ or } w-7=0[/tex]
Solve for each case:
[tex]\displaystyle w = -\frac{8}{3} \text{ or } w=7[/tex]
Since the width cannot be negative, we can ignore the first solution.
Therefore, the width of the rectangle is seven centimeters.
Thus, the length will be:
[tex]\ell = 3(7) - 13 = 8[/tex]
Thus, the dimensions of the rectangle are 8 by 7 centimeters.
