You are building a right-angled triangular flower garden along a stream as shown in the figure.

the fencing of the left border costs $8 per foot, while the fencing of the lower border costs $2 per foot. (no fencing is required along the river.) you want to spend $400 and enclose as much area as possible. what are the dimensions of your garden, and what area does it enclose? [the area of a right-triangle is given by
a = xy/2.]

Given conditions:

Total spending limits = 400$

Left border cost = 8$ per foot

Lower border cost = 2$ per foot

Area of the garden = A = xy/2

Required answers:

Dimension of the garden

&

Area enclosed by the garden

Solution:

Let suppose

The left border of the garden = y

The lower border of the garden = x

According to given conditions;

400 = 2x + 8y (1)

Where left side (400) represents the total spending limits on the garden and right side represents the total cost on the left and lower border.

Solving the equation; 

2x = 400 – 8y (2)

x = (400 – 8y) / 2

x = 200 – 4y (3)

As area is given as

A = xy/2 (4)

Putting equation (3) in equation (4), we get

A = ((200 – 4y) * y) /2

A = (200y – 4 y^2 ) / 2

Or

A = 100y – 2y^2 (5)

To get the maximum garden area, take the derivative of equation (5) and make it equal to “0”;

dA/dy = d/dy (100y – 2y^2)

dA/dy = 100 – 4y

Now

100 – 4y = 0

4y = 100

y = 25

Putting value of y = 25 in equation in equation (1);

400 = 2x + 8(25)

400 = 2x + 200

2x = 400 – 200

2x = 200

x = 100

Thus the dimensions of the garden are as follow ;

Left border length = 25 foot

Lower border length = 100 foot

While the right border length can be calculated by Pythagoras theorem, as the garden is in shape of a right angled triangle;

Let the right side = z

Then z = square root of (x^2 + y^2)

z = square root of ((100)^2 +(25)^2)

z = square root of (10000 + 625 )

z = square root of ( 10625 )

z = 103 foot (approximately)

now the area of the garden is given as ;

A = xy/2

Putting values;

A = (100 * 25) / 2

So area of the garden is calculated as;

A = 1250 foot^2

x = 17

step-by-step explanation:

1st step: isolate x.

24 - 3x = -27

-24         -24

-3x   =   -51

2nd step: divide.

-3x     =   -51

  3             3

  x       =     17

hope this