need help with this problem please

Question

A rectangular garden with area of 30 sq. meters is surrounded by a grass border 1 m. wide on the east and west side but 2 m. wide on the north and south side. What dimensions of the garden minimize the combined area of garden and borders?

Answer 1

Alright, let's break this problem down.

First, draw a picture of garden with border to understand the later area sums/calculations.

To get the the main point, we want to minimize the total area of the garden and border to save as much room as possible. In calculus jargon, this means the rate of change of the total area (A_T) with respect to both sides of the inner garden will be equal.

dA_T/dS₁ = dA_T/dS₂ [The area isn't changing with respect to the North/South side any more than the East/West side of the garden, so this is the goal, now we just need to find A_T]

A_T = A_g + A_b [Total area equals the sum of garden and border area combined]

Can you figure it out from there? (Hint, find area of garden and area of border in terms of your given distances and the unknown distances, then at the end, use the area of the garden to determine the minimum side lengths you want)

If you're still stumped, follow these next guidelines:

S₁ = North/South sides of garden

S₂ = East/West sides of garden

d₁ = Constant length of border along east/west flanks (1 [m])

d₂ = Constant length of border along north/south flanks (2[m])

1. A_g = S₁S₂
2. A_b = 4d₁d₂ + 2d₂S₁ + 2d₁S₂
3. A_T = S₁S₂ + 4d₁d₂ + 2d₂S₁ + 2d₁S₂

Next, take the derivative of (3) for each variable side of garden...

dA_T/dS₁ = dA_T/dS₂

This should lead you to S₂ = S₁ - 2 [m]

A_g = 30 [m²]

= S₂S₁

= (S₁ - 2 [m])(S₁)

= S₁² - 2S₁ - 30 = 0 [Quadratic formula solves for S₁]

Solve for S₁ and S₂ and those are your minimized lengths for the garden.