You are building a right-angled triangular flower garden along a stream as shown

Question

You are building a right-angled triangular flower garden along a stream as shown in the figure. (The borders can be in any directions as long as they are at right angles as shown.) The fencing of the left border costs $5 per foot, while the fencing of the lower border costs $1 per foot. (No fencing is required along the river.) You want to spend $140 and enclose as much area as possible. What are the dimensions of your garden, and what area does it enclose? HINT [The area of a right-triangle is given by A = xy/2.] left border Incorrect: Your answer is incorrect. ft bottom border Incorrect: Your answer is incorrect. ft garden area ft2

Figure,

A right triangle with its longest side taken up by the river

Explanation / Answer

given the cost per foot on the left side is $5 per foot

For, lower border costs $1 per foot

and we need to spend 140$

let

x = length of the left side triangle

y = length of the lower border

since we need to spend 140$

so, total amount spend while constructing left side is = 5*x = 5x

total amount spend while constructing lower side = 1*y = y

so total amount spends on constructing is = 5x+y

but given total amount is 140$

5x+y= 140

possible values of x and y

(note: the taken values should satisfy the right angle triangles)

X =[5,27] and y = [5,135]

so all the possible values will cost the same amount = 140$

but we need the maximum area

solution 1:

we need to check the values

so of all the values, x= 14 and y = 70

will give maximum area

solution -2:

it is a technique, that if the same amount is spent on constructing both the sides it will give the maximum area

so 5x=70 , x= 14 and y=70

so finally

maximum area is

= xy/2 = 14*70 / 2 = 490 sq.ft

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