1(a) a rectangular plot is bounded at the back by a river. No fence is needed along the river and there is to be an opening in front of about 24 meters. If the cost of fencing in front is $1500 per meter and the cost of fencing on the sides is $1000 per meter, find the dimensions of the largest plot which can be fenced for $300 000.

Question

(b) a manufacturer wants to design an open box having a square base and a surface area of 108 cm^2. What dimension will produce a box with maximum volume?

Answer 1

Let each of the sides be x m
let the length along the front be y m

so fencing needed
= 2x + (y-24) , ----- you said we needed an opening of 24 m at the front

1500(y-24) + 1000(2x) = 300000
1500y - 36000 + 2000x = 300000
2000x + 1500y = 336000
20x + 15y = 3360
4x + 3y = 672
y = (672 - 4x)/3 = 224 - (4/3)x

Area = A = xy
= x(224 - (4/3)x) = 224x - (4/3)x^2

dA/dx = 224 - (8/3)x
= 0 for a max of A
(8/3)x = 224
8x = 672
x = 84
then y = 224 - (4/3)(84 = 112

The rectangle should have a frontal length of 112 m and 2 sides of 84 m each.

b) let the base be x by x
let the height be y

SA = x^2 + 4xy
x^2 + 4xy = 108
4xy = 108 - x^2
y = (108-x^2)/(4x)

volume = x^2 y
V = x^2(108 - x^2)/(4x) = 27x - (1/4)x^3
dV/dx = 27 - (3/4)x^2 = 0 for a max/min of V
(3/4)x^2 = 27
x^2 = 36
x = 6 , rejecting the negative answer
then y = (108-36)/24 = 3
max volume = x^2 y = 108 cm^3

-- which is not a coincidence, by the way.

Answer 2

Thanks ReinY

Answer 3

The second term in the expansion of (1-x)(1+2x)Λ4 is 19x. Find the value of n