Respuesta :
Answer:
The dimensions of the rectangle are 7 inches by 5 inches;
[tex]\begin{gathered} \text{length = 7 inches} \\ \text{width = 5 inches} \end{gathered}[/tex]Explanation:
Given that the length of a rectangle is 8 inches shorter than three times its width.
Let x represent its length and w represent its width.
[tex]x=3w-8\text{ -----1}[/tex]If the area of the rectangle is 35 square inches;
[tex]xw=35\text{ ----2}[/tex]Making w the subject of formula in equation 2;
[tex]w=\frac{35}{x}[/tex]substituting into equation 1;
[tex]\begin{gathered} x=3w-8\text{ -----1} \\ x=3(\frac{35}{x})-8 \\ \text{multiply through by x;} \\ x^2=105-8x \\ x^2+8x-105=0 \end{gathered}[/tex]Solving for x in the quadratic equation;
[tex]\begin{gathered} (x+15)(x-7)=0 \\ x=-15 \\ \text{and} \\ x=7 \end{gathered}[/tex]Since length cannot be negative;
[tex]x=7[/tex]substituting to equation 2;
[tex]\begin{gathered} xw=35 \\ 7w=35 \\ w=\frac{35}{7} \\ w=5 \end{gathered}[/tex]Therefore, the dimensions of the rectangle are 7 inches by 5 inches;
[tex]\begin{gathered} \text{length = 7 inches} \\ \text{width = 5 inches} \end{gathered}[/tex]