17. 1/2 points | Previous Answers SCalcET8 4.7.037 My Notes Ask Your Teacher A p

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17. 1/2 points | Previous Answers SCalcET8 4.7.037 My Notes Ask Your Teacher A piece of wire 24 m long is cut into two pieces. One piece is bent into a square and the other is bent into an equilateral triangle (a) How much wire should be used for the square in order to maximize the total area? 24 (b) How much wire should be used for the square in order to minimize the total area? Enhanced Feedback Please try again and draw a diagram. Keep in mind that the area of a square with edge a is As a2 and the area of an equilateral triangle with edge b is AVS 4 Let x be the perimeter of the square, which means x = 4a, and y be the perimeter of the triangle, which means y = 3b. Find a relationship between x and y, considering that the wire's length is a constant and x +y. Rewrite the total area A As At as a function of one variable. Use calculus to find the edges of the square and the triangle that maximize the area; then find the edges that minimize the area Need Help?Read It Watch It Talk to a Tutor Submit Answer Save Progress Practice Another Version 18. 1 points SCalcET8 47 054 MI My Notes Ask Your Teacher Find the equation of the line through the point (3, 5) that cuts off the least area from the first quadrant. Need Help? Read ItMaster itTalk to a Tutor

Explanation / Answer

Minimize the area :
Take x to be the perimeter used for the triangle
And take 24 - x to be the perimeter used for the square

Side lngth of triangle is :
s1 = x/3
And area of triangle At = sqrt3/4 * x^2/9

Sidelength of square is :
s2 = (24 - x)/4
And area o the square is : (24 - x)/4 * (24 - x)/4
As = (24 - x)^2/16

Total area is :
A = At + As

A = x^2 * sqrt(3)/36 + (24 - x)^2/16

Now, derivin' :
dZ/dx = 2x*sqrt(3)/36 + 1/16 * 2(24 - x) * (-1) = 0

xsqrt(3)/18 - (24-x)/8 = 0

xsqrt(3)/18 = (24-x)/8

Crossmultiply :
8x*sqrt3 = 18(24-x)

8x*sqrt3 = 432 - 18x

x = 432/(18 + 8sqrt(3))

How much was used for the squre?
24 - x indeedio!

Now, find that yo!

24 - 432/(18 + 8sqrt(3))

(432 + 192sqrt(3) - 432) / (18 + 8sqrt3)

192sqrt(3) / (18 + 8sqrt(3)) ------> ANS

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