Answer to Question #191917 in Calculus for Juvy

Question

Solve the following problems involving optimization.

1. A rectangular is to be fenced off along the bank river where no fence is required along the bank. If the material for the fence costs 60 pesos per running foot for the two ends and 90 pesos per running foot for the side parallel to the river, find the dimensions of the field of the largest possible area that can be enclosed with 27,000 pesos worth of fence.

2. Find a pair of non-negative number that have a product of 162 and minimize the sum of two times the first number and second number with closed bounded internal of [1,10].

Answer 1

1.) The equation for the given condition can be formed as,

$27000 = 60x+60x+90y$

$120x+90y = 27000$

$y = \dfrac{27000-120x}{90}$

Let the area be A,

$A = xy$

$A = x(\dfrac{27000-120x}{90})$

$A'(x) = 300-2.66x = 0$

$x = \dfrac{300}{2.666} = 112.78$

$y = \dfrac{27000-120\times 112.78}{90}$

$y = 149.62$

2.) $xy = 162$

$S = x + 2y$

$S = x+2(\dfrac{162}{x})$

$S'(x) = 1-\dfrac{324}{x^2} = 0$

$x^2 = 324$

$x = 18$

$y = \dfrac{162}{18} = 9$