144k views
2 votes
What is the shortest possible perimeter for an arrangement with an area of 15 square feet? Please get me the answer as soon as possible!!!!!!!!!!!

User Landon G
by
7.6k points

1 Answer

5 votes

Answer:

The shortest possible perimeter is:
4√(15)

Explanation:

Given


A = 15 --- area

Required

Find the shortest possible perimeter

Area is calculated as:


A=l * b

This gives:


l * b = 15

Make l the subject


l=(15)/(b)

Perimeter is calculated as:


P =2(l + b)

Substitute
l=(15)/(b)


P =2((15)/(b) + b})

Rewrite as:


P =2(15b^(-1) + b)

Differentiate and minimize;


P' =2(-15b^(-2) + 1)

Minimize by equating P' to 0


2(-15b^(-2) + 1) = 0

Divide through by 2


-15b^(-2) + 1 = 0


-15b^(-2) = -1

Divide through by -1


15b^(-2) = 1

Rewrite as:


(15)/(b^2) = 1

Solve for b^2


b^2 = (15)/(1)


b^2 = 15

Solve for b


b = \sqrt{15

Recall that:
P =2((15)/(b) + b})


P = 2 * ((15)/(√(15)) + √(15))


P = 2 * (√(15) + √(15))


P = 2 * (2√(15))


P = 4√(15)

The shortest possible perimeter is:
4√(15)

User Danny Bravo
by
7.7k points