Question

Justin wants to use 188 ft of fencing to fence off the greatest possible rectangular area for a garden. What dimensions should he use? What will be the area of the garden?

how do I solve this?

Answer 1

length =47 ft

width = 47 ft

Area=2209 square feet

Explanation:

Hello

let's remember the rectangular area equation

A=length (l)* width( w)

the perimeter of that area is given by

P=2l+2w

step 1

Let

x=length of the greatest posible area

y=width of the greatest posible area

so

`A_(max) =x*y(equation\ (1)`

the perimeter of that area is

`P=2x+2y`

step 2

justin use 188ft,hence

`P=2x+2y=188\\2x+2y=188\\subtract\ 2x\ in\ each\ side\\2x+2y-2x=188-2x\\2y=188-2x\\\\divide\ each\ by\ 2\\(2y)/(2)=(188-2x)/(2) \\y=94-x`

y=94-x equation (2)

step 2

Now replace (2) in (1)

`A_(max) =x*y\\A_(max) =x*(94-x)\\A_(max) =94x-x^(2)`

Now, he have the area as a function of x

`A_(max) =94x-x^(2)`

derive to find the maxims of the function,by doing A(x)' = 0

`A_(max) =94x-x^(2)\\A' =94-2x \\A' =94-2x\\94-2x=0\\x=47`

x=47 (equation 3)

to figure out if is a maxim verify

`A(x)'' > 0?`

`A' =94-2x\\A'' =94-2\\92 >0`

so, x=94

effectively is a maxim

Step 3

replace (3) in (2)

`A_(max) =x*y\\A_(max) =94x-x^(2)equation(4)\\replacing x=47 in (4)\\A_(max)=94*47-47^(2)\\ A_(max)=2209\ ft^(2)`

Step 4

now, replace (3) in (1) to find y

`y=94-x\\y=94-47\\y=47`

step 5

answer

the dimensions should be

length =47 ft

width = 47 ft

the greatest area possible is 47ft*47 ft = 2209 square feet

Have a great day

Answer 2

the greatest rectangular area would actually be a square

so 188/4 = 47 feet per side

so 47ft x 47ft

area = 47^2 = 2209 square feet