Question

Can you guys pls help me with this math question

Answer 1

Answer:

Dimensions:  150 m x 150 m

Area:  22,500m²

Step-by-step explanation:

Given information:

• Rectangular field
• Total amount of fencing = 600m
• All 4 sides of the field need to be fenced

Let $x$ = width of the field

Let $y$ = length of the field

Create two equations from the given information:

  Area of field:   $A= xy$

  Perimeter of fence:   $2(x + y) = 600$

Rearrange the equation for the perimeter of the fence to make y the subject:

$\begin{aligned} \implies 2(x + y) & = 600\\ x+y & = 300\\y & = 300-x\end{aligned}$

Substitute this into the equation for Area:

$\begin{aligned}\implies A & = xy\\& = x(300-x)\\& = 300x-x^2 \end{aligned}$

To find the value of x that will make the area a maximum, differentiate A with respect to x:

$\begin{aligned}A & =300x-x^2\\\implies \dfrac{dA}{dx}& =300-2x\end{aligned}$

Set it to zero and solve for x:

$\begin{aligned}\dfrac{dA}{dx} & =0\\ \implies 300-2x & =0 \\ x & = 150 \end{aligned}$

Substitute the found value of x into the original equation for the perimeter and solve for y:

$\begin{aligned}2(x + y) & = 600\\\implies 2(150)+2y & = 600\\2y & = 300\\y & = 150\end{aligned}$

Therefore, the dimensions that will give Tanya the maximum area are:

150 m x 150 m

The maximum area is:

$\begin{aligned}\implies \sf Area_{max} & = xy\\& = 150 \cdot 150\\& = 22500\: \sf m^2 \end{aligned}$