Question

A garden has an area of 336 f?. Its length is 5 ft more than its width. What are the dimensions of the garden?The width of the garden is(Simplify your answers.)and the length of the garden is

Answer 1

To answer this question, we can proceed as follows:

1. We know that the area of a rectangle is given by:

`A_(rectangle)=lw`

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:

`\begin{gathered} 336=(x+5)x \\ x(x+5)=336 \end{gathered}`

4. We have to multiply the terms on the left side of the equation as follows:

`\begin{gathered} x(x+5)=336 \\ x(x)+x(5)=336 \\ x^2+5x=336 \end{gathered}`

5. Now we need to subtract 336 from both sides of the equation:

`\begin{gathered} x^2+5x-336=336-336 \\ x^2+5x-336=0 \end{gathered}`

6. We have a quadratic equation, and we can solve it using the quadratic formula as follows:

`\begin{gathered} x=(-b\pm√(b^2-4ac))/(2a) \\ \\ ax^2+bx+c=0 \end{gathered}`

7. From the resulting quadratic function, we have:

`a=1,b=5,c=-336`

Then, we have:

8. From the answer, we have two possible solutions here:

`\begin{gathered} x=(-5+37)/(2)=(32)/(2)\Rightarrow x=16 \\ \\ x=(-5-37)/(2)=(-42)/(2)\Rightarrow x=-21 \end{gathered}`

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:

`\begin{gathered} w=x\Rightarrow w=16ft \\ l=x+5\Rightarrow l=16ft+5ft=21ft \end{gathered}`

In summary, we have that:

The width of the garden is 16ft, and the length of the garden is 21ft.