# Calculus

A square piece of canvas measuring 10 ft. by 10 ft. is to be used to make a pup tent with open ends as shown. How long should the center pole be if the volume covered by the tent is to be maximum?

1. 👍 0
2. 👎 0
3. 👁 161
1. I guess we are not allowed to cut this canvas up so the tent will be ten feet long and the opening at the end will be an equilateral triangle with slopes(hypotenuse of right triangle bounded by center pole h, slope, and ground g, of 5 ft.

Maximize the area of that triangle
A = .5 g h
dA/dh= .5 g dh/dh + .5 h dg/dh
= .5 g + .5 h dg/dh
= 0 for max or min

g^2 + h^2 = 5^2 = 25
g^2 = 25 - h^2
g = sqrt (25-h^2) = (25-h^2)^.5

dg/dh = .5(25-h^2)^-.5 (-2h)
= - h (25-h^2)^-.5
so
0 = .5(25-h^2)^.5 - .5h^2 (25-h^2)^-.5

(25-h^2)^.5 = h^2 /(25-h^2)
25 - h^2 = h^2
2 h^2 = 25
h^2 = 5^2 /2
h = 5/sqrt 2 = 5 sqrt 2/2 = 3.54 ft

1. 👍 0
2. 👎 0

## Similar Questions

1. ### Calculus

Squares with sides of length x are cut out of each corner of a rectangular piece of cardboard measuring 3 ft by 4 ft. The resulting piece of cardboard is then folded into a box without a lid. Find the volume of the largest box

asked by Michelle on November 5, 2015
2. ### Calculus

a piece of wire 12 ft. long is cut into two pieces. one piece is made into a circle and the other piece is made into a square. Let the piece of length x be formed into a circle. allow x to equal 0 or 12, so all the wire is used

asked by Tara on July 3, 2012
3. ### Math

A rectangular piece of cardboard measuring 12 cm by 18 cm is to be made into a box with an open top by cutting equal size squares from each corner and folding up the sides. Let x represent the length of a side of each square in

asked by NR on March 23, 2017
4. ### calculus

A piece of wire 18 m long is cut into two pieces. One piece is bent into a square and the other is bent into a circle. (a) How much wire should be used for the square in order to maximize the total area?

asked by Anonymous on November 3, 2013
5. ### math

It costs \$0.50 per square yard to waterproof canvas. What will it cost to waterproof a canvas truck cover that is 15'x24'?

asked by gee on June 26, 2015
1. ### math

A box with a square base and no top is to be made from a square piece of cardboard by cutting 4-in. squares from each corner and folding up the sides, as shown in the figure. The box is to hold 324 in3. How big a piece of

asked by G on January 24, 2015
2. ### Algebra Homework, HELP!

1) A rectangle has an area of 297 square inches and a width which is three inches less than four times the length. Find the length and the width of the rectangle. 2) Let x represent the length of the shorter leg in a right

asked by Preston on October 14, 2013
3. ### math help

A piece of rectangular paper measures 8x by 12x. If a circle with a radius of 3x is cut out of the middle of the paper, what will be the area of the remaining piece of paper in terms of π? 3. A rectangular gazebo measuring 7x by

asked by XenaGonzalez on March 21, 2016
4. ### Algebra 0110 College

1) A rectangle has perimeter 38 feet and an area of 78 square feet. Find the dimensions of the rectangle assuming the width is smaller than the length. L= W= 2) A rectangular canvas picture measures 9 inches by 7 inches. The

asked by Quince on October 15, 2013
5. ### Math Geometry

Marisol is painting on a piece of canvas that has an area of 180 square inches. The length of the Painting is 1 1/4 times the width. What are the dimensions of the painting this is for elementary school so make it simple

asked by Donald on March 7, 2014
6. ### Calculus

Squares with sides of length x are cut out of each corner of a rectangular piece of cardboard measuring 3 ft by 4 ft. The resulting piece of cardboard is then folded into a box without a lid. Find the volume of the largest box

asked by Allison on November 4, 2012