To answer this question, we can proceed as follows:
1. We know that the area of a rectangle is given by:
[tex]A_{rectangle}=lw[/tex]2. And we have that:
• The ,length, is 5ft more than its width ---> x + 5.
,• The ,width, is x.
,• The ,area of the garden, is 336 square feet.
3. Now, we have that:
[tex]\begin{gathered} 336=(x+5)x \\ x(x+5)=336 \end{gathered}[/tex]4. We have to multiply the terms on the left side of the equation as follows:
[tex]\begin{gathered} x(x+5)=336 \\ x(x)+x(5)=336 \\ x^2+5x=336 \end{gathered}[/tex]5. Now we need to subtract 336 from both sides of the equation:
[tex]\begin{gathered} x^2+5x-336=336-336 \\ x^2+5x-336=0 \end{gathered}[/tex]6. We have a quadratic equation, and we can solve it using the quadratic formula as follows:
[tex]\begin{gathered} x=\frac{-b\pm\sqrt{b^2-4ac}}{2a} \\ \\ ax^2+bx+c=0 \end{gathered}[/tex]7. From the resulting quadratic function, we have:
[tex]a=1,b=5,c=-336[/tex]Then, we have:
[tex]\begin{gathered} x=\frac{-b\pm\sqrt{b^2-4ac}}{2a} \\ \\ x=\frac{-5\operatorname{\pm}\sqrt{5^2-4(1)(-336)}}{2(1)} \\ \\ x=\frac{-5\pm\sqrt{25+1344}}{2} \\ \\ x=\frac{-5\pm\sqrt{1369}}{2} \\ \\ x=\frac{-5\pm37}{2} \end{gathered}[/tex]8. From the answer, we have two possible solutions here:
[tex]\begin{gathered} x=\frac{-5+37}{2}=\frac{32}{2}\Rightarrow x=16 \\ \\ x=\frac{-5-37}{2}=\frac{-42}{2}\Rightarrow x=-21 \end{gathered}[/tex]9. Since the value of x = -21 is meaningless to this answer - the values for length or width cannot be negative, then the value for x = 16.
10. Now, to find the values for the width and the length, we have:
[tex]\begin{gathered} w=x\Rightarrow w=16ft \\ l=x+5\Rightarrow l=16ft+5ft=21ft \end{gathered}[/tex]In summary, we have that:
The width of the garden is 16ft, and the length of the garden is 21ft.