Question

Can you guys pls help me with this math question

Answer 1

Answer:

Dimensions:  125 m x 250 m

Area:  31,250 m²

Step-by-step explanation:

Given information:

• Total amount of fencing = 500m
• Only 3 sides of the land need to be fenced

First, let us assume that the land is rectangular in shape.

Let $y$ = length of the side opposite the river

Let $x$ = length of the other 2 sides of the land

Therefore, we can create two equations from the given information:

  Area of land:   $A= xy$

  Perimeter of fence:   $2x + y = 500$

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

$\begin{aligned} \implies 2x + y & = 500\\ y & = 500-2x \end{aligned}$

Substitute this into the equation for Area:

$\begin{aligned}\implies A & = xy\\& = x(500-2x)\\& = 500x-2x^2 \end{aligned}$

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

$\begin{aligned}A & =500x-2x^2\\\implies \dfrac{dA}{dx}& =500-4x\end{aligned}$

Set it to zero and solve for x:

$\begin{aligned}\dfrac{dA}{dx} & =0\\ \implies 500-4x & =0 \\ x & = 125 \end{aligned}$

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

$\begin{aligned}2x + y & = 500\\\implies 2(125)+y & = 500\\250+y & = 500\\y & = 250\end{aligned}$

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

125 m x 250 m (where 250 m is the side opposite the river)

The maximum area is:

$\begin{aligned}\implies \sf Area_{max} & = xy\\& = 125 \cdot 250\\& = 31250\: \sf m^2 \end{aligned}$