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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.]

Verified Answer :

zeshan

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

 

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